What happens to Two-Face at the end of the movie?
Batman knocks Two-Face off the building where the villain had taken Gordon's family. He is immobile on the ground, good eye closed, and is presumably dead. Gordon and Batman decide to pin Two-Face's crimes on Batman in order to preserve Dent's image as the "White Knight," giving the citizens of Gotham hope. A memorial service is held for Dent, where Gordon can be seen making a speech.
Note: There is speculation among audiences that "the funeral" is faked and that Harvey Dent survived the fall. See this FAQ entry.
If Harvey Dent is really dead, then wouldn't this violate Batman's code of not killing his enemies?
There is no definitive answer. Some say Batman broke his code; others say he did not.
One interpretation: Batman did break his one rule in order to save Gordon's son, which is what the Joker prophesied earlier in the film by saying, "Tonight you're going to break your one rule." However, The Joker's initial plan was to get Batman to kill the Joker himself; showing that everyone is corruptible. Batman beat him and showed that the Joker couldn't force him to break his rule. Later, Batman chooses to break his rule by killing Two-Face. The moral is that Batman won't be forced into doing anything by a maniac. He chooses to do the right thing even though people might hate him for it.
Another interpretation: Batman pushed Two-Face away to save Gordon's son, but Two-Face wouldn't let the boy go. Batman's choice was to save Gordon's son and let Two-Face fall or vice-versa. Batman chose to let Two-Face fall; but he didn't kill him.
What hope do fans have that Two-Face will be brought back for the third film?
1) The coin Two-Face flips in the air (to determine the fate of Gordon's son) lands on the "good" side after his fall. However, surviving the fall can also be looked at as a coin flip. Batman lives; Two-Face, "not so lucky".
2) Gordon and Batman never explicitly state that Two-Face is dead. Even Gordon says "Whatever chance you gave us of saving the city dies with Harvey's reputation.." Which is only saying that Dent's reputation has been destroyed and not Dent himself.
3) Gordon has already covered up Dent's murders and his escape from the hospital. These cover-ups could foreshadow the even bigger cover-up of Dent's supposed death.
4) At some point, it is believed, to get back into the good graces of the majority of the city, Batman needs to be cleared of his crimes, and the only one who can effectively clear him is Dent.
5) The height that Batman and Dent fall from doesn't appear to be extremely high. Batman is knocked out and battered by his landing, but still fine enough to be able to run a couple of minutes later (thought admittedly he is wearing body armour). Batman drops Maroni off a rooftop, saying he knows the fall from that height isn't enough to kill a man. In addition, Batman and Rachel survive a much larger fall from Wayne's penthouse earlier in the film. It's just as plausible that the fall may have just knocked Dent into a coma.
6) Aaron Eckhart said he would like to be in the next installment of Batman, suggesting that he also thinks Dent might still be alive.
7) Writer Jonathan Nolan said in an interview that the fall killed Harvey Dent, but producer Emma Thomas has stated that Dent's fate was left ambiguous. [source needed]
8) Earlier in the film, the people of Gotham are allowed to believe that Gordon has died, a lie that proves to be part of an elaborate cover-up. The event could be a foreshadowing of Dent's resurrection.
9) Many viewers claim that they have seen Dent breath at least 2 or 3 times after his fall. It is highly unlikely that this is the unintentional result of a crucial direction oversight, as Christopher Nolan is widely known for his close attention to detail and highly choreographed cuts in his movies. In addition, such an obvious onscreen goof could have been easily solved with a more closely-directed retake, or else edited out in post-production.