The Great Gatsby offers many examples of irony. A number of these moments have already been pointed out in the posts above.
The novel opens with one bit of irony that is often commented on. Nick describes himself on the opening page of the novel as someone "inclined to reserve all judgments" yet in the same paragraph he presents an evaluation (a judgment) of the many young men that have taken opportunities to tell him their stories. These stories are, according to Nick, "plagiaristic and marred by obvious suppressions." While Nick is not exactly condemning these young men and their stories, he is offering a judgment and situating himself as a narrator that will comment subjectively on his narrative.
Another example of irony in Gatsby is the scene where Daisy cries over Gatsby's shirts.
“They’re such beautiful shirts,” she sobbed.… “It makes me sad because I’ve never seen such—such beautiful shirts before.”
This episode takes place in Chapter 5 and presents an irony within the romance of Daisy and Gatsby. The shirts are symbolic, functioning as an emblem of Gatsby's success and the material comfort that his success brings. Such success is ironic in two ways in this scene.
First, Gatsby and Daisy were once in love but did not marry because Gatsby was too poor. Acquiring wealth, he has now become Daisy's ideal match. Second, the love affair between Gatsby and Daisy is repeatedly contextualized - by Gatsby - as an affair of the heart. There is a sense that this affair represents almost a platonic ideal, a perfect romance. Yet the "heart" of the romance is as materialistic as it is emotional.
Affection and love seem to take a back seat to Daisy's impression of Gatsby's business success (and to Gatsby's desire to prove himself materially worthy of Daisy's love).