The ultimate shortcoming of Fortunato that seals his fate is the folly of pride. His lofty estimates of his skill in wine connoisseurship is what allows Montresor, his friend and eventual murderer, to lure Fortunato into his home with the temptation of trying the rare vintage of Amontillado. It is Fortunato's pride that allows him to be blinded to Montresor's machinations against him, which he had been plotting even while pretending to be his friend. Presumably, it is that same obstructive pride that led Fortunato to repeatedly insult Montresor in the first place, which in turn sparked Montresor's need for revenge.
Additional social critiques could be oriented around Fortunato's gluttonous consumption of wine, to the point where he was intoxicated and thus unsuspecting and ill-prepared for Montresor's actions against him, and more fundamentally, that Fortunato essentially fell into the death trap because of his desire to drink.
Other deeper moral quandaries arise with Montresor's motivation based on perceived "insults," which he never describes further from that point. His comments towards Fortunato in the catacombs imply that both he and his once great family have fallen in status, that he is no longer happy or successful, and that the Montresor family is no longer in possession of their elevated social position. This disparity is highlighted by Fortunato being a Mason, while Montresor is not. In this light, the story could be critiquing the lengths Montresor went to to destroy Fortunato, literally "the fortunate one," out of jealousy.