This kinda goes without saying, but Heath Ledger's death totally changed the plans for the Joker. Rather than just the defining glue of one of the best movies of the 21st Century, he could have been the overarching villain of the greatest movie franchise ever.
Now, Christopher Nolan famously doesn't plan ahead, never looking beyond the release of the movie he's currently working on. This enables his undivided focus on the matter at hand and leads to totally complete films that don't invest inordinate time into setting up a sequel (something far too prevalent in modern blockbuster cinema). It also means any plans for a third Batman film were, at the time of Ledger's death, vague at best. It took him a few years, and the production of a whole other film, to even consider making The Dark Knight Rises.
Still, it's clear from the second film of the trilogy that any subsequent movie, whether or not directed by Nolan, would have featured a returning Joker. Nothing in The Dark Knight was changed to accommodate Ledger's passing, meaning his ambiguous capture in the final moments was purposely keeping the door open. What exactly his role in events would have been is obviously unknown (it likely wouldn't have been anything as expansive as Bane's hostage crisis), but it'd have probably kicked off with him manipulating events from Arkham (not dissimilar to the rumoured plans for Jared Leto's take in Suicide Squad).