Although hist fate isn't addressed in this film, it was brought up in the Dark Knight Rises: the Official Novelization by Greg Cox, which is a movie tie-in novel. It's not a good book but it is the proper novel version of the film and there's a small paragraph that mentions the Joker is, according to rumor, Arkham Asylum's only patient and everyone else had been moved to Blackgate Prison but it's unsure if he's still there of if he may have escaped.

You can read the excerpt here.