I think the only reason I could think of presently is the fact that Nolan wanted to preserve the memory of the character created by Heath Ledger and decided to end it with Dark Knight. Any reference to him in the movie would've created a scene portraying him since he isn't alive the creators would've refrained from making any reference to it which could've resulted in negative publicity / reception because of his iconic portrayal in Dark Knight.
Given the Fact "Spoilers below"
We see Liam Neeson's character speaking to Bruce again even though he was killed off in the First movie, had Heath Ledger been alive I think we could've seen a different take on the whole movie with Joker probably playing another major role (Because of Heath Ledger's dialogue in Dark knight "I think you and I are destined to this forever")
As for what could've happened to Dark knight. The last scene where he is shown is the scene where is hanging on the building towards the end of Dark knight where the swat team enters pointing their gun at him.
My assumption is that they could've taken him to Arkham because of his instability as the reason, where he could've been mourning over the failure of his plan and also since there was no activity about Batman finding no inspiration he could've died at Arkham or broken free and leave the city for good. or He could've been killed by the SWAT team because he killed many cops, blew up hospitals. I don't think he was put along with Dent's prisoners because if that was the case he would've been brought back out during the war sequences towards the end, so I think he died at some point during the peaceful times in Gotham and if he had left Gotham definitely the war at Gotham would've prompted him to come back to join in on the fun.