Viola's disguise (Dramatic Irony)
Most of the film is motored by a central dramatic irony: the fact that we know that Viola is a girl in disguise, while the characters miraculously have no idea and take her "Sebastian" at face value. So many of the comedic premises and so much of the plot revolves around the fact that the viewer knows something that the characters do not—something that would completely change the course of events if it was revealed.
Sebastian returning a day early (Dramatic Irony)
Viola only wants to pose as Sebastian until the game against Cornwall, but when Sebastian returns from London a day early, she has no idea. He shows up at school and goes to sleep without anyone even noticing. Thus, the viewer knows that Sebastian has returned before Viola does.
Viola telling her mother she's going to learn about becoming a debutante (Situational Irony)
As an excuse to get away from her mother so she can successfully pose as Sebastian, Viola tells her prissy mother that she is going to stay with her dad so she can learn the ways of a debutante from Monique. Viola's mother is overjoyed to hear that Viola is finally taking femininity more seriously. It is then very ironic that, in the next scene, Viola sets to work dressing up as a boy, so she can go play soccer on a boys team.
Olivia is only flirting with Duke to make "Sebastian" jealous (Dramatic Irony)
When Viola, dressed as Sebastian, doesn't seem very interested in Olivia, Olivia hatches a plot to get her attention. She starts flirting with Duke, who is in love with her, in hopes that it will make Viola jealous. It does make Viola jealous—but not because she has a crush on Olivia, because she has a crush on Duke. In this scenario there are several layers of dramatic irony. The viewer knows that Viola has a crush on Duke, even though Duke and Olivia do not. The viewer also knows that Olivia is just flirting with Duke for more attention from Viola, which Duke and Viola do not know.