This is something that's been on my mind for quite a while, now... Remember, the one gangster that Batman threatened? "A fall from this height wouldn't kill me." Foreshadowing, almost. Unless Dent broke his neck on the fall, my answer would have to be a strong 'no'.
My best guess would probably have to be he was sent to Arkham as Two-Face. The funeral was staged for the people. Harvey Dent was dead. Dead to the world, society: Gotham. His drop most likely did not kill him. Broke a few bones, maybe, threw him into unconsciousness. Nothing fatal.
The police department had probably wanted to give him a hero's funeral. They wanted the citizens to remember him as Harvey Dent: Gotham's White Knight. Not Two-Face, madman. "Because that’s what needs to happen. Because sometimes the truth isn’t good enough, sometimes people deserve more, sometimes people deserve to have their faith rewarded."
Just a theory. A very plausible theory.