We know that Montresor is insane because he kills a man for no reason.
To be insane is to not have full use of your mental faculties. In other words, a person thinks things that a sane person never would. He or she may see or hear things that or not there, or imagine things. This person will do things that a sane person would never do, like kill a person.
We can tell Montresor is insane because he decides to kill a man, Fortunato, because of a ridiculous reason. He says that Fortunato has insulted him and has to die because of that.
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 I gave utterance to a threat. At length I would be avenged …
Most people do not kill others because of an insult, especially a minor one, and certainly not with this much planning. It is not normal, or sane. Actually, killing another person is not a normal or sane thing to do in general. Planning a meticulous murder and wanting to get away with it is a thing that some people do, and it is not something in and of itself that makes a person insane. What makes Montresor insane is the fact that he found something minor as the motive and meticulously planned a murder from it.
We know that the “thousand injuries” were minor because first of all it is hyperbole (gross exaggeration) and second of all, if Fortunato had really done something to Montresor there is no way he would have gone into the catacombs with him! If you have done real insult to someone and there is a chance he might be planning revenge, you don’t go underground with him.
Montresor asks Fortunato to look at a special cask of wine with him.
I said to him—“My dear Fortunato, you are luckily met. How remarkably well you are looking to-day. But I have received a pipe of what passes for Amontillado, and I have my doubts.”
Fortunato is surprised, especially since Montresor wants him to look at the wine “in the middle of the carnival,” but he goes along with it because of his pride. Montresor tells him he needs his opinion, and even says he might get another man’s opinion, and Fortunato is having none of that.
Montresor goes through with killing Fortunato and feels no remorse until he is dead. This again shows that he is insane. He is more concerned with getting away with the murder.