From the beginning of the story, we are confronted with a narrator who seems to be in complete charge of his faculties--he plans and carries out a perfect murder, an exercise in logic--and, yet, our first encounter with him also leads us to suspect that he might be imbalanced:
THE thousand injuries of Fortunato I had borne as I best could, but when he ventured upon insult I vowed revenge. You, who so well know the nature of my soul, will not suppose, however, that gave utterance to a threat.
The problem for the reader, of course, is that Montresor never explains how he could put up with a "thousand injuries" but becomes a murderer based on "insult." Our inability, throughout the story, to understand the insult creates a tension between our view of Montresor as either a terribly wounded sane man or a man who has been pushed to insanity by a real or imagined insult.
Clearly, Montresor's plan to destroy Fortunato is an exercise in logic and careful planning. For example, Montresor has a flawless understanding of Fortunato, particularly his prideful personality, and he traps Fortunato with his love of rare wine and his boastful expertise:
He had a weak point --this Fortunato --although in other regards he was a man to be respected and even feared. He prided himself on his connoisseurship in wine
Montresor, at this point, seems to be both logical and rational in his ability to exploit Fortunato's weaknesses. Yet, at the same time, even though we might be impressed by the lengths to which Montresor goes in order to get Fortunato into the catacombs under Montresor's palazzo, we are conscious that this clever plan is leading to Fortunato's death.
As the men descend deeper into the catacombs, Fortunato comments on being surrounded by Montresor's family (the bones of Montresor's ancestors line the walls) and asks about Montresor's coat-of-arms:
"A huge human foot d'or, in a field azure; the foot crushes a serpent rampant whose fangs are imbedded in the heel."
"And the motto?"
"Nemo me impune lacessit."
We realize at this point that Montresor's actions so far are reflected by his family motto: he is the foot of gold who is crushing a serpent (Fortunato) with his heel. More important, though, is the motto, which loosely translates to "No one harms me with without retribution." Here, one can argue that perhaps Montresor, the descendant of a violent family, is simply acting in accord with his family traditions--if someone harms you, revenge yourself upon them and, by the way, don't get caught.
The reader's problem, though, is that we can never ascertain whether Montresor is completely sane or insane because we are never told what Montresor's justification is. As long as we can never understand the difference between a "thousand injuries" and one insult, we have no choice but to conclude that Montresor is, if not completely insane, at least temporarily insane as a result of the insult.
There are many reasons why a reader might suspect that Montresor, the narrator of "The Cask of Amontillado," is insane to a greater or lesser degree.
To begin at the end, it seems like insanity, not to want to kill somebody, but to kill him in such a horrible fashion. Montresor chains Fortunato to the rock wall and leaves him there for fifty years, during which time he undoubtedly enjoys imagining his victim turning into a skeleton in the moldering rags of his jester's costume.
Montresor hates Fortunato enough to want to kill him in this horrible way, yet he continuously refers to him as "my friend," "my good friend," and "my poor friend." And he seems to mean it sincerely.
When Montresor gets Fortunato down the stairs into his vault he acts in a zany manner like a lunatic. He claims to be a Mason and daringly shows his victim the trowel he has concealed under his cloak.
When Fortunato asks about his coat of arms, Montresor makes up a wild, garish image of a golden foot crushing a snake which has its fangs in the foot's heel. This is probably the pure invention of a deranged man caught up in his own imagination. He also invents a motto to go with the golden foot which is too appropriate. He is virtually warning Fortunato that he is in extreme danger.
Finally, Montresor's words and actions while he is walling up his victim sound somewhat insane. He screams. He echoes Fortunato's words when the horrified man says, "For the love of God, Montresor!" Fortunato believes Montresor is insane and is trying to humor him.
A case could be made that Montresor is insane. Someone might even create a paper with the thesis that Montresor is deliberately creating an insanity defense in case something goes wrong and he is tried either for murder or attempted murder.