JKR never elaborated on what happened to the corpse, in the books, or any of the interviews based on my searches (admittedly, I didn't check Pottermore).
All we know was that the body was dead, and never mentioned again:
"...Nevertheless, I was as powerless as the weakest creature alive, and without the means to help myself...for I had no body, and every spell that might have helped me required the use of a wand. . . ." (GoF)
There are plenty of fan theories, but none of them have canon confirmation.
The three most popular options are:
It was destroyed in the blast that destroyed Potters' home
It was found by the Ministry and somehow disposed of (circumstantial evidence of this option is that people celebrated Voldemort's death the next day - and they had to SOMEHOW have known that he was killed, eh? I doubt they would have simply announced his death on a random guess without seeing the body, and there were no living witnesses of the death aside from Harry Potter and in-hiding-assumed-dead-Peter-Pettigrew).
Wormtail took care of it (hidden, buried or vanished), at the same time he retrieved Voldemort's wand. (This one seems unlikely, as taking care of a body would require time he didn't have, especially if in a disguise of a rat).