In "The Impossible Astronaut", River Song said, "A Time Lord's body is a miracle. Even a dead one. There are whole empires out there who'd rip this world apart for just one cell. We can't leave him here. Or anywhere." However, that doesn't really explain why.
In any case, River is known to lie (as is the Doctor) & it's possible that she was lying on this occasion. As we learned in "The Wedding of River Song", it was not the Doctor's body. It was the Teselecta imitating the Doctor's body. The supposed cremation served to let the Teselecta sink to the bottom of the lake & then to walk away unobserved. Burning the "body" almost immediately also meant that nobody had time to examine it & to find it wasn't real.
We don't know if River was telling the truth or if she was giving a cover story to explain why the "body" had to be disposed of in that way & why it had to be done so quickly -- or even doing a bit of both.