One good example of irony comes when Fortunato coughs in the catacomb from the damp air; Montresor seems to worry about his ill-health, and asks him to return to the surface:
"We will go back; you will be ill, and I cannot be responsible. Besides, there is Luchesi--"
"Enough," he said; "the cough's a mere nothing; it will not kill me. I shall not die of a cough."
"True -- true," I replied...
(Poe, "The Cask of Amontillado," eNotes eText)
Montresor shows that he is not even slightly worried that Fortunato will die of a cough. This is because Montresor is assured of his own role in Fortunato's death, and knows precisely what will actually kill him. The irony is that, in claiming concern for Fortunato's health, Montresor actually shows his personal assurance that the plan will play out as he intends, and is not worried that anything will interfere. Montresor is also not afraid to show hints of his plan, even displaying the trowel he will use to brick Fortunato into his tomb.
One good example of irony in "The Cask of Amontillado" is in Montresor's continually referring to Fortunato as his friend. This is the man he hates so bitterly that he has been planning for years to murder him in a horrible fashion. He calls Fortunato "my friend" and repeatedly describes him as "my friend," "my good friend," and "my poor friend." We understand that Montresor has forced himself to think of Fortunato as his friend for several reasons. For one thing, he wants Fortunato to believe that he regards him as his good friend. Also, Montresor is thinking far ahead. He knows that there will be a great inquiry when it is generally realized that Fortunato has vanished off the face of the earth. He wants to be above suspicion. Nobody would suspect that such a good friend would have any connection with a crime against Fortunato. People will be talking about this disappearance for years, and Montresor, as a very good friend, will have to keep asking about his friend until the investigation and the speculation finally die down. So Montresor has been practicing thinking of Fortunato as his best friend even while he hates him. Every time he uses the word "friend" in dialogue or narration it is ironic. A complex character like Montresor can condition himself to like a man he hates.