Herman Melville does not specify what happens to Moby Dick after the whale destroys the Pequod and all the whaling boats, killing everyone but the narrator Ishmael. Captain Ahab is caught in a harpoon line and is dragged down to his death by the whale. There seems no reason to think that Moby Dick would have died, although he might be burdened with harpoons and entangled in harpoon lines and might even be dragging Ahab with him for some time. However, the harpoons should work themselves loose eventually, and that would rid Moby Dick of the lines and of Captain Ahab. According to the e-notes Summary of the final chapters of the novel:
Ahab’s boat is the last one left. He once again throws his harpoon and makes fast to Moby Dick. The line runs afoul, and as Ahab stoops to clear it, the line catches him around the neck. He is silently pulled from the boat, down into the depths by Moby Dick.
The white whale, Moby Dick, lives in the end.
After searching for months, the Pequod finally spots the white whale and gives chase. This chase lasts for days as the ship closes in on the whale. Ultimately, the whale is harpooned (by Ahab) and turns to face its pursuer.
Attacking the ship, Moby Dick damages and sinks the Pequod killing everyone but Ishmael. Ishmael survives because he finds Quequeg's coffin, which has risen from the whirlpool created by the sinking ship. With this coffin, Ishmael is able to survive until he is rescued.
Ahab dies along with the rest of the crew, dragged down by the whale into the sea.
Your question is HUGE and totally central to the story's themes, because in order to find out what "happens" to Moby-Dick in the end, we have to analyze what Moby-Dick is in the beginning!
Let's do a simple 1-2 break down of what Moby-Dick is and what happens to him (or her) at the end of the novel!
1) The simplest answer is, Moby-Dick is a whale. An animal. That's it. Although it's a basic approach to a complex novel it's not bad start. In fact the other title of Melville's masterpiece is Moby Dick; or The Whale. So, Ahab finally slew the whale, of course he died as well. They probably both sunk to the ocean floor: a fitting end for a great sea captain and Ancient sea creature! However, if you want to make this argument a little more interesting, The Epilogue was added to the novel after the initial printing of Moby-Dick, because readers didn't know who was narrating the story if it never explained how Ishmael survived in the first place!(1)
2) A common approach is to make Moby-Dick a symbol for God or Satan. Throughout the novel there are references to hell, heaven, devils, angels, and countless quotes from the bible. Heck, Ahab's last words are "From Hell's heart I stab at thee.." Also throughout the novel there are accounts that Moby-Dick is seen in different parts of the world simultaneously! An omnipresent being!
Also one of the last lines from the books are "...The flag Ahab went down with his ship, which Like Satan, would not sink to hell till she had dragged a living part of heaven along with her..." (Melville 452). Perhaps the "corpse-less" end to the novel is justified if we subscribe to this symbolization of God/Satan. If you feel confident and you're writing a long paper, don't be afraid to bring in Nietzsche's "God is dead" arguments to supplement this kind of analysis.
Whatever way we go about representing the whale, ultimately there isn't any confirmation that the whale is dead or alive. Ishmael doesn't see a lifeless white body. For a story that's pretty ambiguous to begin with, this ending should be expected. If you're writing a paper, you may be able to tie this mysterious/ambiguous ending to the painting Ishmael tries to decipher while in the Spouter-Inn in chapter 3.