He should live, definitely. Be locked up in Arkham or whatever, but still alive and a threat.
Maybe carry over to a third movie, where it's a big free-for-all with Joker and Two-Face.
I'd prefer these new films
not take the Burton and Schumacher approach and feel like they have to toss in every villain from the comics because, let's face it, they get kinda silly...I don't think there's a way to do Riddler and Penguin without the camp (or the whiff of camp and "ewwww, this is kinda lame").
Joker and Two-Face are legit hardcases and bad men. Let Batman tangle with those two in this movie and, if necessary, one more.
But I'll go so far as to say this: I don't
need a third movie either. There's no law saying "The Dark Knight" can't be a completely satisfying two-hour or so experience, with a defeated - but still alive - Joker, and a triumphant Batman heading out into the night to do his thing.
I just have a thing about third movies in a franchise...ugh.
Two works fine: introduce a story, establish a vibe and "rules". Then make ONE sequel, expand on it a bit, toss some unexpected elements into the mix ("Luke, I am your father", etc.), resolve some things and leave a few things hanging for fans and movie buffs to talk about for years to come ("Star Wars", "X Men", Burton's Batman, Raimi's "Spider-Man" and "The Godfather"
all should've stopped after their second one; leave on a high note, leave 'em wanting more...but nooooooooo, gotta go and make those turdfest third entries don't they?).
Trilogies, schmilogies...so trumped-up, meaningless and "expected" nowadays, isn't it?