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 I gave utterance to a threat.At length I would be avenged; this was a point definitely, settled—but the very definitiveness with which it was resolved precluded the idea of risk. I must not only punish but punish with impunity. 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.

This is the thinking of a sick mind - planning revenge for an insult! He even tells us he planned this "at length" and didn't merely threaten his friend. Worse, he is going to punish "with impunity." We can actually imagine him smiling his evil smile:

I continued, as was my wont to smile in his face, and he did not perceive that my smile now was at the thought of his immolation. 

As he slyly traps Fortunato into coming with him to taste his wine, we see evidence of his twisted mind in everything he says because we know what he has planned. Finally, when he almost finishes plastering up the wall, he says:

I reapproached the wall; I replied to the yells of him who clamoured. I re-echoed, I aided, I surpassed them in volume and in strength. I did this, and the clamourer grew still.

At this point, we can imagine him brandishing his sword around his head, screaming, yelling, trying to yell louder than the man he is burying alive. ACKGHGGG!!!

You can read the story here on eNotes.

In Edgar Allan Poe's short story, "The Cask of Amontillado," the narrator and protagonist, Montresor, is found to be mentally unbalanced or deranged to some extent.  Though Poe never directly states that Montresor is insane, that fact is made evident by the same character's actions and speech.  While the information given in the opening paragraph is enough to suggest to the reader that Montresor is not well, the contents of the rest of the story offer many pieces of evidence that tell the reader that Montresor's mental illness is a certainty.

In the first paragraph of the story, Montresor divulges his requirements for revenge and states that, should his plan fail to result in meeting each of the prerequisites, he would not have been vendicated at all.  The extreme nature of his views on his vendetta indicate that Montresor is "off."

As Fortunato is led to the scene of his imprisonment and, ultimately, death, Montresor frequently mentions his concern for Fortunato's health while also luring him to continue onward. 

"Come," said I, with decision.  "we will go back; your health is precious.  You are rich, respected, admired, beloved; you are happy, as once I was.  You are a man to be missed.  For me it is no matter.  We will go back; you will be ill, and I cannot be responsible.  Besides, there is Luchesi--"

Perhaps the most telling evidence of Montresor's deranged state occurs after he becomes startled by the screams coming from Fortunato as he is being entombed.  After his initial fright, which leads Montresor to attempt to stab Fortunato, Montresor calms himself, reapproaches the partially sealed compartment, and screams back at Fortunato so loudly that Fortunato's cries are overpowered.

...I reapproached the wall; I replied to the yells of him who clamored.  I reechoed, I aided, I surpassed them in volume and in strength...

Clearly, these are not the actions of a mentally stable person.

Few people kill over a verbal insult.  Poe uses the old adage, “Sticks and stones may hurt my bones, but words will never kill me,” in reverse.  In Montresor’s world, words are deadly.  Insanity drives the main character and narrator Montresor to murder Fortunato who has insulted his family a thousand times [a hyperbole at best]. There have been times in history when insults would have led to a duel with swords or pistols.  But this killing is different!  It is ruthless and cruel beyond measure.

In Edgar Allan Poe's short story “The Cask of Amontillado,” Montresor speaks to his audience in a soliloquy, relating a fateful meeting between himself and his secret enemy Fortunato during the carnival season.  Enraged beyond reason because of a careless insult made to his family by Fortunato, the narrator sets in motion his plan to murder the unsuspecting wine connoisseur.

Montresor exhibits his madness by his detailed planning of the murder. Luring his enemy into the tombs of his ancestors which also serve as wine cellars, Montresor tells his enemy that he has procured a valuable barrel of Amontillado and that he wants Fortunato to taste it. Unfortunately for Fortunato, there is no Amontillado hidden deep beneath the ground. Instead, Montresor chains him up to the wall in the catacombs and buries him alive.

I busied myself among the pile of bones of which I have before spoken.  Throwing them aside, I soon uncovered a quantity of building stone and mortar.  With the material and with the aid of my trowel, I began vigorously to wall up the entrance of the niche.

Montresor has become a socio-pathic killer.  His behavior is deplorable and despicable though he believes himself clever because of the success of his plan.   The murder of Fortunato demonstrates Montresor’s callous disregard for moral restraint.  These restraints keep normal people from committing violence against another person.

Through his madness, Montresor is able to put an invisible mask of civility and friendship.  His caring façade enables him to lure Fortunato into the catacombs and on to his death.  He masks his insanity by playing the part of an inept wine connoisseur to Fortunato’s arrogant expertise. 

His ability to act the friend to his hated enemy when his anger has become unmanageable follows  the logic of William Blake’s poem “The Poison Tree.”  By not expressing his anger, it has grown so much  that he infuriated and willing to kill to gain vengeance. From words to a horrific murder takes many insane steps.

Another ironic aspect of the story comes from the variation in the characters’ appearances.  Montresor plays the rational well-turned individual while Fortunato is dressed like the fool. The dress of each character, therefore, seems as if it should be switched to fit the persona and motives of each character.  But, by masking his true intentions, Montresor is able to hide his insanity through his appearance.  

Poe uses verbal irony to accentuate the hardness of Montresor‘s heart. When the ignoramus Fortunato has a coughing spell, Montresor feigns concern offering him more wine to keep Fortunato drunk in order to easily chain him.  His ironic statements also add to the reader’s knowledge that this is a man who will kill and with impunity.

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.

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.

The narrator is certainly an unreliable one. The story is told from Montresor, the narrator's, point of view. At the opening of the story Montresor claims that Fortunato has committed "a thousand insults" against him, but he never names what was done. When Fortunato makes his way into the story he thinks the two to be friends so Montresor could have the insults worked up in his mind. At the close of the story as he is burying Fortunato alive, which isn't exactly a sane act to begin with, he remains quite calm even when Fortunato begins to realize what is happening to him. A reader can deduce from the act of burying him alive that Montresor is in fact insane. He does have a moment of feeling guilty, but he attributes the feeling to the dampness of the catacombs, which again could point toward Montresor being insane.

"My heart grew sick - on account of the dampness of the catacombs."