According to the description of the events given on page 254 in chapter 14, Fire and Ice , Smaug's bones remained where they had fallen. As far as I can recall, their status wasn't mentioned again in the post-Hobbit Lord Of The Rings novels.
They removed northward higher up the shore; for ever after they had a
dread of the water where the dragon lay. He would never again return
to his golden bed, but was stretched cold as stone, twisted upon the
floor of the shallows. There for ages his huge bones could be seen in
calm weather amid the ruined piles of the old town. But few dared to
dive into the shivering water or recover the precious stones that fell
from his rotting carcase.
The Hobbit- p.254