Discuss the obvious irony in this story: what is the role of luck or fortune?

There is a dramatic irony in this story in that we, the readers, are aware of something that one of the characters, Fortunato, is tragically not aware of, namely that his apparent friend, Montresor, means to do him harm. Most of the suspense and tension of the story derives from this irony, which also increases our sensitivity to how luckless the ironically named Fortunato really is.
There is also situational irony in the second half of the story when Montresor takes Fortunato to his wine cellar, which happens to require passage through "the catacombs of the Montresors." Passing through these burial places, where "piled skeletons" line the walls, Fortunato is oblivious to Montresor's intentions because he, Fortunato, is so keen to taste the famous Amontillado. One could argue here that Fortunato is naive rather than luckless. Indeed, the setting seems to rather ominously, and rather obviously foreshadow his fate. He foolishly expects to arrive in a wine cellar, but instead encounters only a small, "interior crypt or recess, in depth about four feet, in width three, in height six or seven." This recess becomes Fortunato's casket, or coffin. This irony is foreshadowed by the title of the story. While the word "cask" ostensibly refers to a barrel of wine, it also derives from the same root word as the word "casket."
Another irony in the story, which compounds the aforementioned dramatic irony, can be found in the description of Montresor's family crest. The crest depicts a foot stamping on "a serpent rampant whose fangs are imbedded in the heel." This of course echoes what Montresor intends to do to Fortunato. Where the former is the foot, the latter is the serpent. The appropriateness of the symbol also reminds the reader that Montresor's actions were premeditated, and to an extent determined by his sense of family honor.

There's a delicious irony in Fortunato's name. He's not fortunate at all; he's quite the opposite. Perhaps he's been lucky in the past, when he was able to inflict "a thousand injuries" on Montresor—or so Montresor tells us—but his luck has well and truly run out.
Because now Montresor has a great opportunity, an opportunity presented by the carnival, when everyone's out enjoying themselves, to exact his terrible revenge far from the eyes of the madding crowd. Fortunato, the self-proclaimed connoisseur of fine wine, thinks he's in luck when Montresor invites him to sample a glass or three of vintage Amontillado. But the wine-tasting is just a cunning ruse to lure Fortunato into a trap, to get him into the catacombs where the wicked Montresor will bury him alive.