Using the phrase "Liar! Liar!" seems to be older than the popular phrase associated with burning trousers. OED provides an early citation of the phrase used twice in a row:
a1607 H. Chettle Trag. Hoffman (1631) sig. I2v Lyer, lyer, licke dish.
I was able to locate the full verse in a later edition of the same book.
The phrase "licke dish," is interesting and personally I don't know how to explain that turn of phrase. However, it is remarkably similar to the phrase "lick spit" seen later in a context that seems relevant to pants on fire:
Meanwhile, use of "Liar! liar!" repeated for emphasis appears reasonably common in the 1800's. A few examples:
'I know nothing of acoustics, Signor. I am at a loss to understand your meaning," said he softly.
'Liar! liar!' shouted I.
But his chosen organs, the Tribune, and other black Republican papers, and the black Republican preachers, have unceasingly denied it, with vulgar epithets of lie! lie! and liar! liar!
'Liar, liar!' shouted the woman, springing forward to clutch the jewel
The article cited by Josh in the comments claims that the full line ("Liar, liar, pants on fire") was cited in print in 1933. The earliest reference I could find was from 1945, but clearly alludes at a historical context around the phrase:
"Liar liar pants on fire, Nose as long as a telephone wire!" sang Fenella to the timeless scornful tune of childhood. "Anyway I'm not black," said Christine. "Anyway I'm not white," said Fenella. "So!"
Overall, this collection of references still seems to suggest that the phrase, in some form or another, was used in verse before any sources I've been able to locate.
While I have no direct evidence of a link, the rhyming line "nose as long as a telephone wire" suggests a possible allusion to the popular and widely translated children's novel The Adventures of Pinocchio (1881-1883) by Carlo Collodi, where, as most readers probably know, the marionette Pinocchio's nose grew each time he told lies.
Following this line of speculation, it seems worth noting that at three points in the novel, Pinocchio is threatened with burning as firewood.
He swam on and on. After a while, he turned around again
and called louder than before:
"Good-by, Master. If you ever need a piece of good dry firewood,