Throughout the film, we the viewers and a few of the characters know Batman's true identity, while the majority of characters have no idea. The question of Batman's identity is, in fact, an integral part of the plot, as Joker places a huge price on Batman revealing his identity to the people of Gotham. The viewer knows that Batman is actually Bruce Wayne, a wealthy businessman, but Harvey Dent, Gordon, Joker, and others have no idea. This creates an instance of dramatic irony.
Harvey says he is Batman (Dramatic Irony)
In a definitive moment, Harvey Dent takes the fall for Batman and lies to the press, saying that he is the masked vigilante. We know that this is not true, but the rest of Gotham takes him at his word and believes Harvey, which creates yet another instance of dramatic irony having to do with Batman's identity.
Rachel's Death (Situational Irony)
The Joker tells Batman that Harvey and Rachel are tied up in separate locations where they will both be killed at the same moment if someone doesn't save them. When Batman arrives at the location where he thinks Rachel is, he finds that Joker has sent him to Harvey's abduction spot instead, a tragic twist that ends up getting Rachel killed.
Joker's Scars (Situational Irony)
The first time the Joker tells one of his victims about the origin of his scars, he appears almost sympathetic, detailing a childhood of abuse that has left him mangled and morally bankrupt. However, later, when he tells the story again, it's completely different. This is an instance of horrifying situational irony, in which the viewer realizes that the Joker changes his story constantly just to manipulate others and amuse himself.
The Dark Knight Questions and Answers
The Question and Answer section for The Dark Knight is a great
resource to ask questions, find answers, and discuss the novel.