LilyPond, 340 characters (shorter than Common Lisp)
'Tis the season! (And rather fitting, given that LilyPond is primarily used to typeset music... One can easily adapt this code to produce sheet music for the song in addition to just the lyrics.)
Adapted from fizzer's solution.
#(map(lambda(x)(format #t"On the ~:R day of Christmas my true love gave to me~{ ~R~:*~[~;~;turtle doves and~;French hens,~;calling birds,~;gold rings,~;geese a-laying,~;swans a-swimming,~;maids a-milking,~;ladies dancing,~;lords a-leaping,~;pipers piping,~;drummers drumming,~]~} a partridge in a pear tree.
"x(iota(1- x)x -1)))(iota 12 1))
Usage: $ lilypond thisfile.ly
This version, in 341 characters, matches fizzer's output exactly, but is three bytes shorter:
#(map(lambda(x)(format #t"On the ~:R day of Christmas my true love gave to me~{
~R ~:*~[~;~;turtle doves and~;french hens,~;calling birds,~;gold rings,~;geese a-laying,~;swans a-swimming,~;maids a-milking,~;ladies dancing,~;lords a-leaping,~;pipers piping,~;drummers drumming,~]~}
a partridge in a pear tree
"x(iota(1- x)x -1)))(iota 12 1))