In Poe's "The Cask of Amontillado," there are several examples of irony. When Montresor explains that revenge is not good enough, but that the victim must know he is being punished, Montresor never explains his actions to his victim, so essentially Fortunato dies without ever knowing why.
I must not only punish but punish with impunity. A wrong is unredressed when retribution overtakes its redresser. It is equally unredressed when the avenger fails to make himself felt as such to him who has done the wrong.
We might find some irony in that Fortunato becomes increasingly more drunk. As a connoisseur of fine wines, and someone who appreciates good wine when it is available, one might think getting him drunk would be more difficult.
For as intelligent as Montresor presents himself to be (and an aristocrat, as well), it's ironic that he does not know what a "Freemason" is—the secret organization known as the Freemasons previously made up of stone masons. Montresor wields his trowel believing that a "mason" is only one that works with stone, as Montresor himself is preparing to do. (This, of course, confuses Fortunato who genuinely is a Freemason.)
He laughed and threw the bottle upwards with a gesticulation I did not understand.
I looked at him in surprise. He repeated the movement—a grotesque one.
“You do not comprehend?” he said.
“Not I,” I replied.
“Then you are not of the brotherhood.”
“You are not of the masons.”
“Yes, yes,” I said; “yes, yes.”
“You? Impossible! A mason?”
“A mason,” I replied.
“A sign,” he said, “a sign.”
“It is this,” I answered, producing from beneath the folds of my roquelaire a trowel.
“You jest,” he exclaimed, recoiling a few paces. “But let us proceed to the Amontillado.”
Another irony may be found in the last words of the story:
In pace requiescat!
This means "Rest in peace!" This is ironic because Montresor seems not to have been able to do just that: he wishes peace (perhaps an ironic one) on Fortunato, but fifty years later, Montresor still tells the tale with precision and mental clarity—we can infer that it has been on his mind, perhaps eating away at him, all this time. To support this assumption, look to Montresor's one moment of hesitation in the story—though he explains it away:
My heart grew sick; it was the dampness of the catacombs that made it so.
This emotional "hiccup" may be something that the author is unable to overcome after the murder.
Poe uses irony to throw off the reader, keeping him/her unbalanced and not sure of Montresor and his intentions. It also serves to make the reader question the narrator of the story: is he reliable or is he insane? As more information comes to light, the story becomes creepier because we assume that Montresor must be insane and is committing not only murder, but he is murdering an innocent man—for the insult may certainly be nothing more than a figment of Montresor's imagination.