The film [that/which] I selected for viewing

The film that I chose for the class to watch is called The Life of Igor.

The film which I chose for the class to watch is called The Life of Igor.

—At the margins, are both correct?

(When I say "at the margins," I am referring to the stipulation in the Chicago Manual of Style Online that
“[It] is more or less okay (and popular among writers of British English) [...] to use ‘which’ restrictively", i.e., Pianos which have a fourth pedal to mute the strings are popular among apartment owners.)”

Or is there some other reason that either "which" or "that" might be disallowed in this case?

NB. If you are troubled by the sentences above for any reason—or are about to post an answer saying something like "It depends on your intended meaning"—please replace "chose" in the sentences with "selected."

The Original Poster's Question

The film that I chose for the class to watch is called The Life of Igor.

The film which I chose for the class to watch is called The Life of Igor.

Both that and which can be used with restrictive relative clauses. A third possibility, is dropping the relative word altogether when the verb in the relative clause has its own Subject.

In the Original Poster's example, the relative clause has a subject, the word I. For this reason the OP could use the following version of the sentence:

The film I chose for the class to watch is called The Life of Igor.

The Original Poster's core question, however, is whether there is any reason not to use which or that here. The answer is, no! There are no situations that I know of where it is wrong to use which for this type of relative clause. However there are several where it is wrong or silly to use that. The original examples don't fall into this bracket, but it is worth examining them anyhow.

When not to use that:

There are some special cases when we can't use that with restrictive relative clauses. The following sentences are fine:

That's the man that you were talking with.
That's the bullet that I was shot with.

However, if we move the preposition to the front of the relative clause, that cannot be the complement of the preposition. We have to use whom or which as appropriate:

That's the man with whom you were talking.
*That's the man with that you were talking. (ungrammatical)
That's the bullet with which I was shot.
*That's the bullet with that I was shot. (ungrammatical)

[Notice that we cannot use who as the complement of a preposition. We have to use whom:

*That's the man with who you were talking. (wrong)

Many writers on this site advise people not to use whom, but this is one situation where you have to!]

Another situation in which it's better not to use that, is after the demonstrative pronoun, 'pointy' that.

^?Do not do that that you know to be wrong. (awkward)

This is not grammatically wrong, but it's a bit awkward, and it can be difficult to read. Lastly we may want to contrast the difference between a person or thing that might have done something. In this case we may want to say who or which. This cannot work with the word that:

Never trust in any people or in any things who, or which, you cannot actually see.
*Never trust in any people or in any things who, or that, you cannot actually see. (ungrammatical)

Conclusion

The Original Poster's examples don't fall into any of these three categories, so either which or that is fine here. However, this is a normal answer for sensible right minded readers. There are several other answers here, which argue that which is incorrect in restrictive relative clauses. To understand why this isn't the case, you may wish to read my answer "In defense of which" below.

In defence of which

The idea that which is not used for restrictive clauses is a myth promulgated in the worst English grammar text-books and style guides ever written. The greatest writers in the English language have continuously used which as a relative pronoun in restrictive relative clauses — as has everyone else too.

Some info from the post Sidney Goldberg on NYT grammar: zero for three, by Geoffrey Pullum, Professor of General Linguistics at the University of Edinburgh, coauthor of the Cambridge Grammar of the English Language (2002), shows how far through various books you would need to go before finding which appearing as a restrictive relative pronoun. The first number given in the list below shows the number of lines in the entire book. The second number shows on which line the author first used which as a relative pronoun in a restrictive relative clause:

A Christmas Carol (Dickens): 1,921 lines, first occurrence on line 217 = 11% of the way through;

Alice in Wonderland (Carroll): 1,618 lines, line 143 = 8%;

Dracula (Stoker): 9,824 lines, line 8 = less than 1%;

Lord Jim (Conrad): 8,045 lines, line 15 = 1%;

Moby Dick (Melville): 10,263 lines, line 103 = 1%;

Wuthering Heights (Bronte): 7,599 lines, line 56 = 0.736%...

Now, I know these authors are idiots who didn't know how to speak English properly, but just how did their editors and publishers get away with it? And all the people who decided to put these books on school syllabi? They, of course, should be shot.

Getting serious again: importantly, as Pullum also shows in the post A Rule Which Will Live in Infamy, there are situations when we, in fact, cannot use that for restrictive clauses and in which we have to use which. (Oh look one of them happened right there. I couldn't for example have written ... "and in that we have to use which"). Here are the three situations Geoff Pullum describes:

The putative ban can’t apply when a preposition precedes the relative pronoun: The town in which she lived is grammatical but *the town in that she lived isn’t.

The supposed rule should be ignored when modifying demonstrative that, because that which you prefer is clearly preferable to ?that that you prefer.

The rule can’t apply to a conjoined which: We must trust the unknown entity who or which created us is grammatical but *We must trust the unknown entity who or that created us isn’t.

Here is the passage in which (Oh no, there I go again with another one!) the famous quote that Pullum is playing with appears:

“Yesterday, December 7, 1941—a date which will live in infamy—the United States of America was suddenly and deliberately attacked by naval and air forces of the Empire of Japan.” That was how President Franklin D. Roosevelt opened his famous infamy speech, 71 years ago. Ignoring the writing handbooks, he opened with a passive construction, which of course is just right for the rhetorical context (America as innocent victim). And he also ignored another bogeyman rule: He introduced a restrictive relative clause with which.

The answer to the OP's question, therefore, is that which can, and sometimes must, be used for restrictive relative clauses. It's not a mistake to do so.

I leave you with Geoff Pullum's last words from the link above:

Grammar snobs trying to show off their linguistic rectitude by playing gotcha with an invented rule that never matched educated usage; copy editors slaving away trying to enforce it; Microsoft Word blindly putting wavy green underlining under every relative which not preceded by a comma. What a senseless waste of time and energy.

Follow the Fowler rule if you want to; it’s up to you. But don’t tell me that it’s crucial or that the best writers respect it. It’s a time-wasting early-20th-century fetish, a bogeyman rule undeserving of the attention of intelligent grownups.

Opinions vary considerably on this one.

That and which are the same

In this Language Log blog (dated 2004), the author calls the distinction between the two nonsense:

...the old nonsense about which being disallowed in what The Cambridge Grammar calls integrated relative clauses (the old-fashioned term is "restrictive" or "defining" relative clauses). Strunk and White perpetuate that myth.

That and which are different

Language Log referred to Strunk and White. E.B. White says (twice!) that these are different. (White expanded Strunk's notes and published the first edition in 1959.)

Nonrestrictive relative clauses are parenthetic.... Commas are therefore needed. (Rule 3, page 3)

That is the defining, or restrictive, pronoun; which the nondefining, or nonrestrictive.

The lawn mower that is broken is in the garage. (Tells which one.)

The lawn mower, which is broken, is in the garage. (Adds a fact about the only mower in question)

(page 59, Elements of Style, 4th Ed.)

Strunk & White would write the clause in the second sentence parenthetically, as:

The film that I chose for the class is called The Life of Igor.

The film, which I chose for the class, is called The Life of Igor.

There would be a slight difference in meaning. In parallel with their lawn mower example, the first sentence informs the reader which film was chosen. This may have been the third of eight films, for example. The second sentence provides more information about the only film you chose for the class.

Conclusion

Language Log would say that the sentences are equivalent. Strunk & White would say that the sentences are saying slightly different things, and that a precise expression on your part would make it clearer to the savvy reader.

My sense is that the language is losing the distinction between restrictive and nonrestrictive clauses. Setting off the nonrestrictive clause with which and commas is beginning to sound formal and archaic.

It is all a question of CONSISTENCY OF REGISTER!

Which is a WORD which/that is more formal than that or the absence of a relative pronoun (both being possible only when the relative pronoun is not the subject of the verb in the relative clause and when this clause is defining/restrictive).

There are also STRUCTURES that are more formal than others as, for instance

the film about which I told you the other day

(preposition preceding the relative pronoun, the 'sort of nonsense up with which Sir Winston Churchill would not put')

versus

the film that I told you about the other day

and

the film ø I told you about the other day

(preposition attached to the verb in both cases).

The point is to be CONSISTENT:

if you use a FORMAL STRUCTURE, then use a FORMAL RELATIVE PRONOUN too!

The film which I told you about the other day

would sound awkward because there would be a DISCREPANCY between an INFORMAL STRUCTURE – preposition attached to the verb rather than preceding the relative pronoun – and a FORMAL RELATIVE PRONOUN, 'which'.

Since the verb in the original poster's sentences does not a have a preposition, there is no risk of discrepancy, which makes a sentence using 'which' as grammatically correct as one using 'that' or dispensing with a relative pronoun altogether… provided the sentence is in keeping, registerwise, with the rest of the text it appears in.

Now, for the reason why 'which','who', and 'whom' are more formal than 'that' or the absence of a relative pronoun, I suppose it is because the latters do not distinguish between people and things; they are less precise.

Reference as to which relative pronouns are formal and which informal:

[google] [http://dictionary.cambridge.org/grammar/british-grammar/relative-pronouns]

Reference for co-variation of relative pronoun and preposition position (away from formal towards informal):

"Briefly, relativization with wh-relative pronouns is giving way to relativization using that or zero. As wh-relativization is strongly associated with prototypical written registers (e.g. news and academic prose), this has to count as another instance of the colloquialization of the written medium. A further parameter – closely connected with this – is the choice between the 'pied piping' construction with a preposed preposition (the project on which I'm working, etc.) and the preposition stranding construction (the project I'm working on, etc.), where the preposition typically occurs in final position in the clause. Again, the tendency is to move away from preposing and toward stranding – perhaps another case where a more learned Romance overlay on English syntax is being undermined by a native Germanic construction more at home in the spoken language."

Whew!