Solution 1:

Christine Ammer, American Heritage Dictionary of Idioms (1997) confirms that "get off" in the sense you're referring to dates to the early 1900s:

get off ... 7. Have the effrontery to do or say something. For example, Where does he get off telling me what to do? [Colloquial; early 1900s]

I suspect that it is related to the expression "on one's high horse." Here is Ammer again, on that idiom:

on one's high horse In an arrogant or condescending manner. For example, When they started talking about music, David got on his high horse and said that classical music was only fit for museums and archives. This expression, alluding to the use of tall horses by high-ranking persons, dates from the late 1700s. Similarly, off one's high horse means "less arrogantly, more humbly," as in I wish she'd get off her high horse and be more friendly. It dates from the early 1900s, but is heard less often today.

According to Ammer, "get off" [meaning #7] and "get off one's high horse" arose at approximately the same time, raising the possibility that the former is a shortening of the latter.

Unfortunately for this theory, however, Chapman & Kipfer, Dictionary of American Slang, third edition (1995), dates "get off one's high horse" only to "by 1928"—a dog's lifetime after the 1913 occurrence of "where do you get off" that you point out. Here's the relevant entry in Chapman & Kipfer:

get off one's high horse v phr by 1928 To stop being haughty and superior; deal informally; =COME OFF one's PERCH [The notion of high horse, "pretentious arrogance," is found by 1716]

On the other hand, that same source states that "get on one's high horse" was current "by 1856," so perhaps people were addressing the question "where do you get off?" to someone who was on his or her high horse, long before they thought to speak of the person afterward as being "off his/her high horse."


UPDATE (8/31/14)

First Occurrences of the Phrase as a Rhetorical Challenge

The Random House Historical Dictionary of American Slang (1994) finds an occurrence of the phrase in George Ade, "The Fable of the Regular Customer and the Copperlined Entertainer," in More Fables (1900). Here is the full paragraph where that instance occurs:

Mr. Byrd's first Move was to take Jim to a Retreat that was full of Statuary and Paintings. It was owned by a gray-haired Beau named Bob, who was a Ringer for a United States Senator, all except the White Coat. Bob wanted to show them a new Tall One called the Mamie Taylor, and after they had Sampled a Couple Jim said it was all right and he believed he would take one. Then he told Bob how much he had taken in the Year before and what his Fixtures cost him, and if anybody didn't think he was Good they could look him up in Bradstreet or Dun, that was all. He said he was a Gentleman, and that no Cheap Skate in a Plug Hat could tell him where to Get Off. This last Remark was intended for an inoffensive Person who had slipped in to get a Rhine Wine and Seltzer, and was pronging about Forty Cents' Worth of Lunch.

Ade returns to the expression "The Fable of the Crafty Love-Maker who Needed a Lady Manager," in The Girl Proposition (1902):

He wore his Chest a few Inches in front of himself and no one could tell him where to get off. Inasmuch as he was a big, husky Good-Looker with all the Manly Accomplishments, he had a Panel Picture of himself leading Miss Blonde into a Flat.

Rosemarie Ostler, Let's Talk Turkey: The Stories Behind America's Favorite Expressions has a fairly lengthy but ultimately inconclusive discussion of "tell someone where to get off," as well.

One of the first occurrences of the challenging question "Where do you get off?" occurs in Alfred Lewis, "The Diary of a New York Policeman," in McClure's Magazine (January 1913):

As a starter I made a meet with Baden. He was rough and high, as I knew he would be. After letting him bullyrag me to his soul's content, I started to leave with—apparently—my tail between my legs.

"Where do you get off," he roared after me—it was perhaps his mildest utterance—"to go givin' me a talk? If you and Muggs go to monkeyin' with me, you'll get it where the baby wore the beads."

An even earlier instance appears in the lyrics to "Personality" "as sung by Eva Tanguay" in The Philistine (March 1911):

Ev'ry time I come to town, the knockers get their hammer,

And they call me pretty names like "Faker" and "Flim-flammer."

Some of them will tell you that I'd rather fight than eat—

Also that, for raising Cain, I've got old Satan beat ;

Actors on the bill say: "Mercy I where does she get off?

On the level, Cull, how does she make them wise guys cough?"

Hark to little Eva, and a tale I'll tell to thee—

Some one wished it on me—why, personality.


An Alternative Origin Theory

Most people who have discussed the two phrases have supposed that the phrase "tell [someone] where to get off" and the question "where does [someone] get off" have a common origin. But Google Books search results suggest the possibility that while the declarative "tell [someone] where to get off" makes sense in the context of a streetcar conductor, the earliest instances of the question "where does [someone] get off" may have an independent origin.

Specifically, the first match for "where does he get off" occurs eleven years before George Ade's first use of the declarative form, and it arises in a peculiar context: a hearing in the California Senate in about hay prices. I'll quote the example at length because the usage is so clearly idiomatic. From Appendix to the Journals of the Senate and Assembly of the Twenty-Eighth Session of the Legislature of the State of California, volume 8 (1889):

MR. HOOK: Was it a necessity of having this ground plowed up by the people that had leased it? A. Was it a necessity?

Yes? A. I look upon it as a necessity to raise all the hay that a man can get.

Q. In the valley? A. Yes, sir; it is impossible almost to get hay in the valley at a reasonable price, and unless there is some hay raised in the valley to regulate the price, there is no telling where you would get off. In 1875 and 1876 I didn't get a ton of hay in the valley for less than $60 a ton, and it came down to $55 and $50; and year before last I didn't buy any hay that I didn't pay $47 50 for, and then couldn't get really enough to do me. I got two bales at a time, and had to turn out some carriage horses to grass. If you take ninety or one hundred head of horses in there, and keep them for seven months, you can figure how much hay it will take; and I undertake to say that right here that it is impossible to get that amount of hay from outside; and you stop raising a little hay in the valley to govern the price, to make a little stand-off with them, and the farmers on the outside, knowing that that hay has to be used, they will pool it as well as any others, railroads or any others, and say, "Here, you can't have our hay for less than $55 or $60 a ton," and I should have to pay it. Now hay forty-five miles—that is, where you have to haul it—forty-five miles from the valley, will sell at home for $20. They don't make anything by hauling that hay and selling it for $45. Hat forty-five miles from the valley is worth $20. Ten months in the year it is worth $20. A four-horse team can't haul but a ton, and a six-horse team will haul a ton and a half, it takes him a week to make the trip—six days—he pays toll over and back. His hay is worth $30 at home—a ton and a half—and when he gets into the valley, where does he get off, after paying freight and expenses? You can see, yourself, that it would be impossible to get it at any price. They don't raise, even then, at the price they would put it at, there is not enough raised within forty-five miles to do the business.

In this excerpt, "where you would get off" seems to mean something like "how you would make [an arrangement] work financially," "how would you get out from under [onerous contract terms]," or simply "how you would benefit [from an arrangement]."The connection to a high horse simply isn't there, and the connection to a railway conductor is remote at best.

This is by no means the only example where "where does one get off" has this distinctive meaning. For instance, from "Prevention of Dealing in Futures," in U.S. Congress, Hearings Before the House Committee on Agriculture, volume 2 (1910), we have this:

Mr. FITCH. Mr. Chairman, I want to say just one word in connection with that term "bucket shop." And in illustration of that, Mr. Chairman, I desire to go back again to Springfield, Ill., to John Smith who is in the grain business. I explained to you how that 10,000 bushels of corn for May delivery ere sold upon the Chicago Exchange; that John Smith was notified to whom it was sold, and the firm who bought it notified the party from whom they bought it, and so forth, making it an actual transaction. I want to show you where those methods differ from what they would have been if John Smith had stepped across the street to one of the small town bucket shops. If he had gone across the street to a bucket shop to make that sale of corn, the method would have been to look at the quotations on the board,the blackboard that these men keep there, and he writes him a ticket at the price, and the transaction ends right there. It would be just the same as if I were to go behind the stand at the race track and lay them 5 to 0 or 5 to 2, and so forth—identically the same proposition.

Now, you may say that if the bucket shop takes these bets, as you call them, where does he get off if the market goes against him? I would say to you that once in a while he finds somebody that is a member of the exchange who will handle some of his business. The member will do it for him under cover and give him a chance to get off, but if a member does it and is found out he never does it the second time; he is expelled.

And from J.E. Rhodes, "An Appeal to Yellow Pine Lumbermen," in Lumber World Review (May 10, 1915):

"WHERE DO YOU GET OFF?"

And, Mr. Timber Owner and Lumber Manufacturer, where do you get off?

You are now facing the absolute necessity of exerting your utmost efforts to restore the lumber industry to what it can easily be, or you will not realize upon your investments anything like the returns that you had anticipated.

And from a letter by M.J. Whittall dated January 27, 1917, to the U.S. House Committee on Interstate and Foreign Commerce, reprinted in Hearings Before the Committee on Interstate and Foreign Commerce of the House of Representatives (1917):

Now, what about the small dealer, not only in New York City but for miles around, where New York City papers are read. Some of these dealers have only floor coverings ; others only floor coverings and furniture. Where do they get off if Mr. Straus succeeds in proving that 15 per cent is a reasonable profit and that everyone should do all their shopping at Macy's?

And from Hugh Kennedy, "Delbart: Timber Cruiser," in Saturday Evening Post (1919):

"I considered it [namely, warning that "they may want to ask you some questions at the other end"] was the least I could do. Purser Tenney, he's a good little man in the main. Bit of an old woman, perhaps, and a little irritable lately; needs a holiday, I should say. It's family trouble, I guess. In this business there's been no reasoning with him."

"Lost his nerve, eh? If they push this thing where does he get off?"

"That's just it! He's under heavy bonds on account of this monthly pay roll. It may go hard with him. He'll lose his job at the least.You can't altogether blame him for being in such a funk. Damned nuisance, I call it, all round!'

And from P.G. Wodehouse, "Aunt Agatha Makes a Bloomer, in Cosmopolitan (October 1922):

"I must say Aunt Agatha, dash it all," I said severely, "I think you have been infernally careless. There's a printed notice in every bedroom in this place saying that there's a safe in the manager's office where jewelry and valuables ought to be placed and you absolutely disregarded it. And what's the result? The first thief who came along simply walked into your room and pinched your perls. And instead of admitting that it was all your fault, you started biting this poor man here in the gizzard. You have been very, very unjust to this poor man."

"Yes, yes," chipped in the poor man.

"And this unfortunate girl, what about her? Where does she get off? You've accused her of stealing the things on absolutely no evidence. I think she would be jolly well advised to bring an action for—for whatever it is and soak you for substantial damages."

Conclusion

Though usage of "where does one get off?" in the sense of what are one's pecuniary self-interest shows up in Google Books search results 22 years before the same question shows up as (arguably) a rhetorical question about someone's gall or nerve, both meanings coexisted for at least a couple of decades after that, before that rhetorical question eventually won out. It's hard to say whether the rhetorical question arose out of the pecuniary self-interest question, but I wouldn't be shocked if it had. Another possibility is that both meanings arose out of some yet earlier ancestor.

In any event I would love to see a qualified historian of the language look into this hitherto underreported sense of "where does one get off?" from 1889.

Solution 2:

My answer from January 2014 accepts the origin date of "by 1928" for the phrase "get off [one’s] high horse" listed in Chapman & Kipfer, Dictionary of American Slang, third edition (1995) and restated as "1928+" in Kipfer & Chapman, Dictionary of American Slang, fourth edition (2007). But a search of newspaper archives yields numerous much earlier instances of related phrases and of the exact wording—and these have led me to rethink my earlier conclusion that any connection between "where do you get off?" and a person dismounting "a high horse" was extremely unlikely.

From the Launcestown [Tasmania] Advertiser (January 11, 1838):

Had the anticipations in which men gladly allowed themselves to indulge, of the results of the new reign, been realised, we should have willingly acknowledged in this call of co-operation a flattering recognition of the community in, if we may so say, its corporate capacity ; as it is, we only see a trimming to circumstances—a dismounting from the high horse, from no love of not riding horses, but from the force of an irresistible expediency. How the community, disappointed and sickened as it is, will answer to the call made upon its exertions, we shall pretend not to say.

From "Star-light," in the [Clarksville, Texas] Standard (May 20, 1848):

Well, we are not about to draw a comparison, or a contrast, between the two [the stars of heaven and the Clarksville Star newspaper], after the style of Brougham or Phillips, but we wish simply to say, that the body of the bright particular Star, the Clarksville Star, loomed above the horizon on yesterday evening, like a blazing Comet and became distinctly and luminously visible in all the rotundity of its full orbed splendor.

However, we believe we shall have to get down from our high horse, and descend to a grade of language far beneath the real merit and importance of the subject, before we shall be able to make ourself understood by common corn raising, beef eating humanity.

From "The Syracuse Doings," in the New-York Tribune (September 14, 1850):

Our City Hunkers will not be pleased that John Van Buren was allowed to rule the roast at Syracuse—that Dickinson was left uncommended—and that there are strong symptoms of his being sold out to secure the election of Seymour & Co. But they are unreasonable, and will have to come down from the high horse they have been riding for some time past and accept the Burners as colaborers and brother Democrats. This being inevitable, they can hardly set about it too soon.

From "The Transit to California," in the New York Daily Times (June 10, 1856), reprinted in the [San Francisco] Daily Alta California (July 26, 1856):

We are not sure as to the prospect of a very prompt alteration in the present schedule, but it will certainly come, and it may be, if Commodore Vanderbilt should consent to alight from his high horse and resume the practical business temper and good sense which usually characterize his management of steamboat jobs, it will be made the profit of his stockholders in the now charterless Nicaragua Transit Company.

From "Save the Union," in the Fremont [Ohio] Journal (January 1, 1858):

The South demands that the Lecompton constitution shall be adopted. Mr. Douglas and the Messenger say no.—The South says it won't stay in the Union if Mr. Douglas and the Messenger don't come down off that high horse. Mr. Douglas and the Messenger reply that they won't do it. Now we ask once mere, with trembling anxiety, what is to become of the Union?

From "Duluth," in the [Duluth, Minnesota] Labor World (September 26, 1896):

The coopers are pushing the boycott on the Imperial mill products. The committee appointed by the Trades Assembly are doing good work and the strong arm of organized labor in Duluth will yet compel Mr. B. C. Church to come off his high horse.

From a review of Rome and the Last Four Popes, in the [Sydney, New South Wales] Empire (June 16, 1858):

It [the book] is very turgid and wordy. We long to see the author got off his high horse and say what he has got, to say in a quiet manner, ^but we are never gratified.

From "Domestic Affairs of a Clergyman," in the London Globe, reprinted in the [Washington, D.C.] Evening Star (September 6, 1858):

According to the evidence of the brother, given in Court, Mr. Vansittart said he wished his wife were dead, and then "she would have no more children, which he detested." But an intimation that the Bishop's attention would be called to the case, brought him down from the high horse to a more I humble attitude, and made him supplicate for mercy. He entreated his wife to forgive him, and she did so.

From "A Battle Royal: Over the Statehood Bill Now Going On at Washington," in the [Nogales, Arizona] Oasis (January 13, 1906):

Before the session of Congress is over the "insurgents" will be glad to get down off from their high horse and vote for a rule demanded by the overwhelming majority of their party conference, Speaker Cannon will show them a few things.

And from "Chicago Street Cars and Freezing Weather," the [Chicago, Illinois] Day Book (January 6, 1912):

When the people use the power they possess as a people, even the arrogant traction interests of Chicago and all their public servants will get off their high horse and do what they could have done long ago—give the people of Chicago transportation facilities worthy of the second city in the United States of America.

It seems to me that these many instances of dismounting from a high horse (figuratively speaking) provide grounds for arguing that George Ade's use (in "The Fable of the Regular Customer and the Copper-Lined Entertainer" [1900]) of "tell him where to get off" in the sense of "whittle him down to size" may very well allude to getting off a high horse—notwithstanding my earlier answer's conclusion to the contrary.

I haven't found any evidence of a direct connection between the two expressions, but "getting off a high horse" antedates "telling someone where to get off," which in turn antedates "where do you get off?" in the relevant sense. "After the fact, therefore because of the fact" is bad logic; but "before the fact" rules out "because of the fact" altogether—and we no longer have a situation where the idiom "tell [someone] where to get off" appears to be older than "get off [someone's] high horse."


Update (July 14, 2020)

I ran Elephind newspaper database searches for "where do you get off?" and allied phrases—and found that the question was posed in multiple instances as a question literally asking someone what location they had in mind for leaving a public conveyance. For example, from "Strangers Yet" in the Saugerties [New York] Weekly Post (April 9, 1891):

“What is the matter?”

“I am afraid to go home,” I said. I was 19 years old, but I felt like a baby.

Where do you get off?” asked the voice.

“At St. John’s place,” I replied, and just then the conductor called out the street, and I pushed my way to the rear platform and stepped off.

From Violet Mitchell, "Miss Tilly's Bonanza," in the [Great Bend, Kansas] Barton County Democrat (May 10, 1894):

"... I don't suppose I'm much better off 'n what you be—but pore is pore —an' a man with a hole in his coat big enough for a cat to crawl through, ain't likely to be no John Jacob As tor. Now be he?"

"No," replied the man, slowly, "he isn't. Where do you get off?" he asked.

"I don't know just where I'll get off," she stammered, "that is I kalkalate to go on to Stillington, if I meet my folks there—that is—"

From "Canadian Simplicity," in the Indianapolis [Indiana] Journal (July 23, 1895):

"You didn't look at my ticket." said I, trying to detain him [the train conductor]."

"Where do you get off?" he Inquired.

"London," said I. "Why, that's a good ways off yet," sail he, smiling. "You've got one, I suppose?"

Many instances similar to these first three appear in subsequent years.

The earliest Elephind match for "Where do you get off?" that might be read as meaning "How do you have the effrontery to do or say something?" appears in "The 'Lulu' in Poker" in the Salt Lake [City, Utah] Herald (April 17, 1899), reprinted from the New York Sun:

"The what?" said the stranger with a bit of warmth in his tone. "What are you giving us? Where do you get off on this, anyhow? You've got a jack high, and mine are fours, which is a walkover," and he started to pull down the pot himself.

As suggesed in my other answer, however, a different slang expression might be the direct source of the expression as used there. Consider these three examples from the 1890s. From "Good or Bad Water?" in the Sacramento [California] Daily Record-Union (August 19, 1893):

Q.—So there is no taxation in all this now?

A.—Not a cent.

Q.—And no increase of water rates?

A.—Not unless you make them yourself. Fish ["the Worthington pump man"] can't do it.

Q.—But where does he get off?

A.—Why, in the sale of water to a growing city for twenty years.

From "Boon to Honolulu: B. F. Dillingham Organizes New Plantation Company," in the [Honolulu, Hawaii] Hawaiian Gazette (December 15, 1896):

"If Hackfeld & Co. are to be the agents for the company, Mr. Dillingham, where do you get off?"

"My interests are largely in the 0. R. & L. Co.; what benefits it helps me. By the consummation of this deal the railway company disposes of a lease of 3,300 acres of land, and Mark Robinson, who has large land interests here, will furnish the land on which the pumping plant and buildings will be located. ..."

And from an untitled item in the Chico [California] Record (September 14, 1897):

The [coal] producers have got their protection to the extent of 30 cents a ton on coal and they have raised the price correspondingly to the consumer. But the man by whose labor it is produced—where does he get off? He asked simply for 15 cents more per ton and he got—cold lead. Is this not an unanswerable showing of the folly and inconsistancy of taxing the consumer to add to the income of the coal trust with the expectation that it will benefit the laborer.

In each of these instances "where do you get off?" amounts to asking "how do you profit from this arrangement?" A fourth instance from the 1890s could be read this way, as well. From "Have Shut Up Shop: Silver Republicans Have Closed Sate Headquarters," in the Salt Lake [City, Utah] Herald (August 13, 1898):

But what of the People's party which decided to fuse with the silver Republicans? Where do they get off at? This question is such a pertinent and grave one that time should be given for thought and the answer will doubtless not be formulated in time for publication before Sunday.

Here, the sense of "where do they get off at?" may be something like "where does that leave them?" or it may be "how do they turn the situation to their benefit?" or it may even be "how do they have the nerve to proceed on their current course?" Of the three options, the first seems to me to be the most likely and the third the least likely. In the "'Lulu' in Poker" story in the same newspaper eight months later, however, the only possible senses of the expression seem to be "how do you benefit from the present situation?" and "how do you have the effrontery to claim that you won?" It may be that the two Salt Lake Herald stories show a transition in the sense of "where do you get off?" from "how do you benefit?" to "how dare you?" At least in those instances, the transition doesn't seem terribly far-fetched.

Solution 3:

Due to how long the expression has been around (as early as 1913), I suspect the phrase "Where do you get off...?" is connected to a blue-collar, or lower-class point of view, associated with commuting home from work on trolley cars, and then later, buses, trains and subways. The implication might be, an offended person is suggesting someone else thinks their views are better than those of others because they daily get off (or exit) the trolley or train (and live) in a well-to-do neighborhood. It's akin to, in an opposite way, of saying someone lives on the "wrong side of the tracks." Living on the "wrong side of the tracks" meant you lived, in terms of real estate values, on the less valuable side of the railroad tracks, that is, the side where the prevailing winds would typically blow the coal dust from the passing trains. It was understood, more well-to-do families could afford to live on the better side of the tracks.