^^ I just assumed that, in keeping with Nolan's more realistic take on the Bat-verse, the Joker had either been executed by the state or was put on death row for his crimes in The Dark Knight
in-between the films, or that he was incarcerated in a maximum-security prison, far outside of Gotham.
Realistically, a lot of jurisdictions would probably be going after the Joker for his crimes. The most obvious, of course, being the City of Gotham (or perhaps Gotham is a combination city-county like San Francisco?). But he also attacked the state's Army National Guard units when he attempted his little "social experiment" on the ferries. (Traditionally, Gotham in the comics has been seen in Batman
fandom as being part of New Jersey, so I'm gonna go with that.) So the New Jersey Attorney General is definitely gonna want a piece of him -- assuming the feds don't want in on the action. (The Army National Guard units in The Dark Knight
look like they're still working for the state and haven't been federalized, though.)
Since New Jersey abolished the death penalty in 2007, my guess is that the Joker (in the Nolan-verse, his real identity was probaby eventually discovered, unlike in the comics -- realistically, it's kind of hard to conceal your identity, especially when you have such prominent facial scars) is sitting pretty in the New Jersey State Prison
during Bane's coup in The Dark Knight Rises
If we place Nolan's Gotham outside of New Jersey -- if we take its similarities to Chicago to mean that it's in Illinois, or its similarities to Manhattan to mean it's in New York State -- then the Joker is probably either in Tamms Correctional Center
or in Sing Sing Correctional Faciity
. Neither state has a death penalty.