“I’ll take you and Tyson to Rockefeller” (Dramatic Irony)
Percy’s mother informs him, “This afternoon we’ll celebrate the end of school. I’ll take you and Tyson to Rockefeller Center—to that skateboard shop you like.” This proves to be an early example of dramatic irony, since soon after, Percy reflects that he wouldn't see home for quite some time. Unbeknownst to any of the characters at the time, the troubles with camp would make going to Rockefeller Center impossible.
“See, Percy? You’ve unlocked your true self!” (Verbal Irony)
Circe is being ironic, or more specifically, sarcastic when she claims that Percy has unlocked his true self. Instead of his true self being the "flawless" vision of himself that he saw earlier in the mirror, Circe believes that his true self is actually a guinea pig. Using verbal irony, Circe is trying to bring Percy down and suggest that he's basically only an animal.
Tyson and Percy Fighting Polyphemus (Situational Irony)
Tyson and Percy fighting Polyphemus is an example of situational irony because of their shared father, Poseidon. In the Odyssey, Polyphemus is able to get revenge on Odysseus by calling out to his father, Poseidon, who curses him to punish him for blinding Polyphemus. This time, however, they are all Poseidon's children, so Polyphemus can no longer use that as a tool to fight.
Kronos's Real Plan (Situational Irony)
The revelation of Kronos's true plan at the end of the novel is an example of situational irony. If we think of situational irony as a kind of plot twist, this may be the biggest example in the novel. While we don't figure out the truth until the last pages, it is clear that Kronos's plan to bring Thalia back from the (almost) dead drove all of the events in the novel.