Fortunato translates, in Italian, to "fortunate one," and he does seem to be very fortunate for the majority of his life, up until the events of this story at least. The narrator, Montresor, in describing his victim, says, "He had a weak point -- this Fortunato -- although in other regards he was a man to be respected and even feared." Thus, life had been kind to this man, really. He has only one weakness and is generally well thought of by the community. He is, indeed, fortunate in this way.
One might expect, then, that the rest of his life would continue in this vein. However, as you point out, his name is ironic, and this means that there must be some discrepancy between expectation and reality. In this case, one would likely conclude from his name that Fortunato would come to a death as fortunate as his life has been. On the contrary, he meets a most unfortunate death, to be walled up alive in an underground crypt until he asphyxiates.
Another irony surrounding Fortunato's name is that, although his end is most unfortunate, he does have that one weakness, and that one weakness is enough for Montresor to exploit and lead the ultimately unfortunate man to his death. If Fortunato weren't so proud of his status as a wine connoisseur, if he didn't believe so strongly that his taste and discernment were so far beyond everyone else's, then Montresor would have had no way to lure him to the catacombs where he had supposed stashed a pipe of Amontillado. So, although the presence of that one flaw was certainly unfortunate for Fortunato, it was extremely fortunate for Montresor!
The irony that lies behind Fortunato's name is that the basic root word of his name is "Fortun" as in fortune, indicating luck, success or prosperity when Fortunato is the actual victim in the story of "The Cask of Amontillado." Fortunato is anything but lucky or fortunate in the story, as he is deceived into trusting Montresor, and he ends up losing his life.
The name Fortunato sounds very much like the word fortunate. Of course, Fortunato is anything but lucky. He is manipulated by Montressor and led to his untimely and horrific death. So the irony lies in the fact that a man whose name would indicate good fortune ends up with the very ill fortune of being buried (or technically walled up) alive--not very fortunate at all.