The story is set during the carnival season around Mardi Gras, just before Lent. It is a festive occasion where people have parties, parades and celebrate life. The irony of the setting is that Montresor is planning on murdering Fortunato, something that Fortunato would definitely not find amusing. But is does set up the contrast between the two characters in the story. One can see Montresor is a very dark, brooding and bitter individual who has remembered some rude remark of Fortunato for so long that he has masterfully planned his death. Fortunato, however, seems to be a rather trusting baffoon interested more in fun and wine than his own safety. Thus the irony of having Fortuanto lose his life during the biggest party of the year.
The carnival was a time of great merry-making, partying, costumes, revelry, drunkenness, and mirth. To be plotting someone's demise amidst all of that is most unexpected. You would expect such a creepy tale of revenge to immediately start off with the more appropriate setting of the cliche "dark and stormy night", not at a happy festival of people. So, the irony exists in the fact that that setting is almost the exact opposite of where they end up, and of what Montresor's dark mind and plans are.
It works well in the fact that first of all, Fortunado is dressed up in a ridiculous costume, with a jingle-bell hat on. It makes his end a bit more pitiful; Montresor pokes and prods in the hole only to hear a few jingles of bells at the end. It is a comical tragedy, almost, to picture Fortunado in that completely unsuited outfit at his death. Also, because of the carnival, all of Montresor's servants "had absconded to make merry in honour of the time", so there were no witnesses to him bringing Fortunado into the catacombs. Very lucky indeed. Then, because of the carnival, Fortunado is drunk, which makes Montresor's plan that much easier; he is more willing to follow, less alarmed, a bit hazy, and doesn't quite get what is going on fast enough. So although the carnival is quite ironic for the morbid tale about to unfold, it works very much in Montresor's favor.
With the gaiety all around him in the setting of the Carnival (Italy's version of Mardi Gras), Montesor plots his evil act of vengeance against the also ironically named Fortunato. Of course, there is a sharp contrast between the backdrop of frolicking, happy drinking, laughter, and revelry and the sinister, duped, drunkard passage of unsuspecting Fortunato as he enters the catacombs of the family of Montesor only to be victimized in a horrific manner. As he wears the harlequin patterns of a court jester, or fool, there lies another irony: Fortunato's clothing reflects his last role in life. Indeed, he is made a fool of as he is lured to his untimely death.
Carnival takes places prior to the start of the Lenten season (similar to Mardi Gras in New Orleans). Lent is when people give up various pleasures - including drinking alcohol. Carnival is a big party. Since Amontillado is a type of fine wine it is ironic that it is used to lure Fortunato to his death. It is also ironic that he never gets to taste the wine!