Where does the phrase "to get on like a house on fire" come from?

The OED says

orig. U.S. like a house on fire (also afire) : as fast as a house would burn; very rapidly or vigorously. Freq. in to get on like a house on fire : (a) to progress rapidly and successfully; (b) (of two people) to establish quickly and maintain a very good relationship.

I think that's the best answer you're going to get.

From Google Answers:

I think this may be lost in the mists of time. The exact Washington Irvin quote mentioned below by pinkfreud appears below, but I have found an earlier reference from 1741 quoted by Thomas Carlyle.

Title: A history of New York, from the beginning of the world to the end of the Dutch dynasty, by Diedrich Knickerbocker.
Author: Irving, Washington, 1783-1859.

Page 473
In proportion, therefore, as a nation, a community, or an individual (possessing the inherent quality of greatness) is involved in perils and misfortunes, In proportion does it rise in grandeur - and even when sinking under calamity, makes, like a house on fire, a more glorious display then ever it did in the fairest period of its prosperity?
http://www.hti.umich.edu/cgi/t/text/text-idx?c=moa;cc=moa;xc=1;xg=1;type=simple;rgn=full%20text;q1=like%20a%20house%20on%20fire;view=reslist;subview=detail;sort=occur;start=1;size=25;didno=ACB2403.0001.001

Title: History of Friedrich the Second, called Frederick the Great
Author: Carlyle, Thomas, 1795-1881.
Publication Info: New York,: Harper & brothers, 1862-1874.

Chap VI
p385
Quoting Burgermeister Spener on 4th Dec 1741
"On the contrary, the love of your burger subjects - that, if you can kindle it, will go on like a house on fire (Ausbruch eines Feueres), and streams of water won't put it out."
http://www.hti.umich.edu/cgi/t/text/text-idx?c=moa;cc=moa;xc=1;xg=1;type=simple;rgn=full%20text;q1=like%20a%20house%20on%20fire;view=reslist;subview=detail;sort=occur;start=1;size=25;didno=ABY8829.0003.001