Some people think he died, and some think he lived. Personally I believe that he died. They showed his funeral during the movie's final montage, plus Aaron Eckhart (the actor who plays Harvey Dent) has said in interviews that Chris Nolan (the director) told him straight up that Dent is dead.
As for the other question (did Batman break his one rule?), I would argue no, he did not. Batman's rule is to not INTENTIONALLY kill anyone. He wasn't TRYING to kill Harvey, he was simply trying to protect Gordon's innocent son. Harvey died in the process unfortunately. It's all about intent, so no, I do not believe that Batman broke his rule...