I guess tehnically, in terms of the law and what Gotham would think, yes - The Batman pushed him off the edge of a building, causing him to fall two or three stories, break his neck, and die upon impact.
But regarding Batman himself, no. He did not break his rule. His rule is not "to not kill" as most people think. It is actually to not execute unnecessarily or in cold blood. He had to tackle Dent to save the boy, and as they all fell, Batman could only hold on to the boy with one arm while holding himself up with the other. He wasn't TRYING to kill Dent.
But regarding Batman himself, no. He did not break his rule. His rule is not "to not kill" as most people think. It is actually to not execute unnecessarily or in cold blood. He had to tackle Dent to save the boy, and as they all fell, Batman could only hold on to the boy with one arm while holding himself up with the other. He wasn't TRYING to kill Dent.