The opening sentence of "The Cask of Amontillado" creates a puzzle that generations of critics and readers have been unable to solve:
The thousand injuries of Fortunato I had borne as best I could; but when he ventured upon insult, I vowed revenge.
Most readers reasonably expect Montresor to describe the insult sufficiently to justify the revenge that drives him to kill Fortunato in a horrific manner. But Montresor simply moves on to explain how he has constructed his revenge, and readers and literary critics are left to puzzle over the "insult." A related issue is that Montresor is the first-person narrator of this tale--everything we see and hear is filtered through the eyes, ears, and mind of Montresor. So, we have a narrator, upon whom we depend for accurate information, who decides to kill a man in a particularly horrific manner because of an insult. We cannot be sure that Montresor is a reliable narrator.
The short answer to your question, then, is that no one is quite sure why Montresor decides to kill Fortunato, except that Montresor thinks the "insult" is sufficient justification for murder. This answer, of course, leaves us in no better position than before because the answer is logically circular: Montresor decides to seek revenge because. That's about as far as readers can go.
Some critics have speculated, among many things, that the insult has something to do with Montresor's family and that Montresor is therefore obligated, as the last remaining Montresor, to defend the Montresor family honor. An intriguing word choice by Montresor may support this theory:
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.
The word immolation is most often used to refer to a ritual sacrifice, not merely a run-of-the-mill murder. Because both Montresor and Fortunato are conscious of their family's status--at one point, Montresor implies that his family's status is reduced--the possibility that Fortunato's insult is directed at the Montresor family, not Montresor himself, becomes important, particularly when we have no other idea why an "insult" would drive Montresor to such revenge.
Both Montresor and Fortunato are from the upper class, and both are from leading aristocratic families in a country--Italy--where loyalty to family is both expected and considered to be a virtue. If, and I realize this is an if, Fortunato has insulted Montresor's family name, it is reasonable to believe that Montresor would feel obligated to take revenge on the family's behalf. This if is supported, I think, by the elaborate discussion of the Montresor family's coat-of-arms: a snake being crushed by a human foot and, in turn, biting the heel of that foot. The snake represents the Montresors, and the foot symbolizes the Fortunato family. One can argue, of course, that the coat-of-arms simply represents the two men's struggle with one another, but coupled with Montresor's use of the word immolation, the elaborate description of the coat-of-arms may point to a family struggle, not a struggle between individuals.
Fortunato, therefore, becomes a sacrfice to the Montresor family's honor, which he has insulted in some way. And Montresor, as the last representative of his family, has lived up to the sentiments expressed on his family's coat-of-arms: No one harms me without suffering himself.