Batman has more than one rule, but his one line that he refuses to cross his one thing that would turn him from a crusader into a villain is that he does not kill. Now only taking Christopher Nolan's Batman into consideration than yes he did break this rule by directly causing Harvey's fall that accidentally killed him.
Let's look at the scene: Batman is shot, Dent is safe and the boys fate is undecided. Batman can't let a coin determine his fate. He lunges at Dent and the boy and they all three go over the edge. Despite his armor he is injured and not at full strength. He is only able to save the boy before he himself falls after Dent. Crashing through more stuff than Dent ( he was pushed far enough out he was clear and may have survived normally if he wasn't in such a badly burned condition) getting more injured on the way down and falling near Dent.
Later on when talking to Gordon: "You either Die a hero or live long enough to see yourself become the villain. I can do those things, because I'm not a hero, like Dent. I killed those people. That's what I can be." ----- "I'm whatever Gotham needs me to be."
It's a great and powerful scene but in it's essence is one man pushing another man off a building and killing him (while saving an innocent boy) and then taking on the blame for not only his death but the deaths caused by Dent as well. Batman is not just telling Gordon that he killed Dent he is saying that he killed those people. Batman feels directly responsible for their death and in Harvey's case is directly responsible. This is the true genius of Nolan's Batman, he not only made him real and plausible he makes him constantly trying to figure out what he's doing, why is he doing it and has he gone to far? He made Batman break his most important rule.
A lot of fans don't want to admit that Batman killed. An accidental killing doesn't count as breaking his rule they say, but to Batman it does count. It will haunt him. The deaths Two-Face caused will haunt him. HE takes the blame for everything that happened. He pushed Two-Face off of a building and he died. The line has been crossed. How will he deal with it? How will he deal with being a criminal? The next movie will definitely deal with this but it is a huge step for Batman.
Nolan has created a realistic Batman. In the first movie we get to see what makes a man become a Batman. What training, what psychological problems, what Gotham's corruption had to do with it. When fighting Ra's al Ghul we start to see a bending of his rule not to kill. Leaving Ra's al Ghul on the train is not killing in his mind because Ra's al Ghul has survived a seemingly unsurvivable situation before plus he's planned meticulously. If Batman can survive it he assumes Ra's al Ghul can as well. Now at the start of the second movie we start to encounter a basic problem with his rule, if Batman doesn't kill why do criminals need to fear him? By the end of the movie Batman has killed and he's taken the blame for more killings. This not only adds to the myth of the Batman but it directly affects the mental state of Batman. All of the second movie we see him wrestling with how far he should push himself, what is his limit, what keeps him from being the Joker? Then he breaks his most important rule and it seems to define him more. Did he plan to kill Two-Face no. Did he directly cause his death yes! He did it in the only allowable way to protect the innocent and by accident but he still killed and broke his rule.
Now we have a whole new Batman going into the third movie. One who has killed. One who has become a villain. Yet he is more sure of himself and what he stands for than ever before.