Poe's use of the unreliable first-person narrator creates several problems in interpretation that have kept "The Cask of Amontillado" a puzzle to readers for generations.
Because Montresor, our first-person narrator, fails to disclose his reason for killing Fortunato in such a horrific manner, readers are left in a moral vacuum--how can readers judge Montresor's actions if they do no understand what, exactly, prompted, those actions? For example, when Montresor tells us that
The thousand injuries of Fortunato I had borne as I best could; but when he ventured upon insult, I vowed revenge.
Montresor's failure to describe the "insult" makes it forever impossible to judge Montresor's actions in a true moral sense--that is, he is a murderer, but is the murder, if not condoned, at least understood? Poe very cleverly--through the use of this unreliable narrator--places Montresor in an ambiguous universe.
Revenge and, perhaps more important, the psychology of revenge, are Poe's themes in the story. In Montresor, we are clearly presented with an intelligent character who understands the implications of his behavior--that is, he understands that killing Fortunato is wrong, so he orchestrates his elaborate plan to make sure he isn't implicated when Fortunato just disappears. For example, he plans the deed at the time of Carnival when everyone is in disguise and difficult to identify; he makes sure his servants have left his pallazzo so that no witnesses intervene; and he makes sure Fortunato is thoroughly drunk. He plans the revenge well.
Montresor's revenge, however, cannot just be accomplished by killing Fortunato. As Montresor states to his confidante in the opening paragraph,
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.
The two important elements of a successful revenge, then, are 1) the person seeking revenge cannot be caught; and 2) the victim must know who is carrying out the revenge. Montresor accomplishes the first by his elaborate plans of time and place, and succeeds in the second when Fortunato recognizes what Montresor is doing at the bottom of the catacombs.
But what has driven Montresor to such a horrific revenge? Again, on the face of things, the reader cannot know the reason because no one knows what the insult is. We can, however, look to Montresor's family motto--Nemo me impune lacessit (No one hurts me without paying for it)--and conclude that Montresor is acting out a familial imperative. The Montresor family, of which Montresor appears to be the last, requires that revenge be carried out.
Montresor may be carrying out a family tradition that is built into his psyche. In other words, as a member of the Montresor family, retribution is a duty that must be accomplished if the family suffers an "insult," so Montresor may not believe he is acting as an individual who has been insulted, but he is convinced that Fortunato's death is necessary to satisfy the Montresor family.
Revenge becomes not just an action but is an integral part of Montresor's psychological make up because he derives from a family for whom retribution is not a choice but a duty.