Two examples of dramatic irony occur (1) in Act II, Scene 2 with Calpurnia's dream, which spurs her to plead with Caesar not to go to the Senate, and (2) in Act III, Scene 1.
Much of the action of Shakespeare's historical plays, such as Julius Caesar, take the form of dramatic irony, since the audience is familiar with much of the historical content in advance, while the characters remain unaware. However, there are specific examples that stand out as this type of irony, in which there is a contradiction between what a character believes and what the reader or audience knows to be true.
1. In Act II, Scene 2, Calpurnia rushes in to Caesar, begging him not to go the Senate. While the audience knows that Brutus has made up his mind to join the conspirators and the assassination is in the making, Caesar feels confident that nothing will happen to him, despite what his wife has seen in her dream, and not knowing what the audience knows.
The gods do this in shame of cowardice.
Caesar should be a beast without a heart
If he should stay at home today for fear.
No, Caesar shall not. (2.2.41-44)
Caesar is convinced that his courage is merely being tested, so he ignores any warnings and goes to the Senate, but the audience knows that it is dangerous for him to go because they have been privy to Brutus's soliloquy
in his garden, as well as the contents of the letter written by Artemidorus.
2. In Act III, Scene 1, as Caesar approaches the Capitol, he sees the soothsayer. Caesar remarks, "The ides of March are come." The Soothsayer replies, "Ay, Caesar, but not gone." Still Caesar ignores his earlier warning in his arrogance. When Artemidorus approaches and tries to hand him the paper that warns Caesar about the conspirators (readers/the audience have knowledge of the content of this paper from Act II, Scene 3). Caesar refuses: "What, is the fellow mad?" (3.1.9) he asks, and Publius pushes Artemidorus away before he can give Caesar his paper.
Because the audience knows that the contents of this paper urge Caesar to beware of Brutus and Cassius, Casca, Cinna, Treboniius, Metellus Cimber Decius Brutus, and Caius Ligarius--the conspirators--they are aware of the dangers Caesar faces while he is unaware of them.