According to official novelization of The Dark Knight Rises, Joker appears to be the sole inmate at Arkham Asylum.
“Blackgate Prison was a maximum-security penitentiary located on one of the smaller islands in Gotham Harbor. Now that the Dent Act had made it all but impossible for the city’s criminals to cop an insanity plea, it (Blackgate Prison) had replaced Arkham Asylum as a preferred location for imprisoning both convicted and suspected felons. The worst of the worst were sent here, except for the Joker, who, rumor had it, was locked away as Arkham’s sole remaining inmate. Or perhaps he had escaped. Nobody was really sure. Not even Selina.”
According to this snippet of interview with Christopher Nolan,
We're not addressing The Joker at all. That is something I felt very strongly about in terms of my relationship with Heath and the experience I went through with him on The Dark Knight".
I didn't want to in any way try and account for a real-life tragedy. That seemed inappropriate to me. We just have a new set of characters and a continuation of Bruce Wayne's story. Not involving The Joker.
So, they only continued with Bane's story and didn't involve Joker at all. They covered the fate of Joker in that novelization.