The protagonist, or villain, in E.A. Poe's "The Tell-Tale Heart" murders the old man because he (the protagonist/narrator) despises the old man's eye: "I made up my mind to take the life of the old man, and thus rid myself of the eye forever." The narrator never speaks of anything wrong the old man did to him; he never claims that he injured or insulted him in any way.
The narrator claims to be sane, numerous times. That said, one does question the narrator's sanity based upon the "fact" that he can hear both heaven and hell (and the fact that he wishes to murder the old man, a man that he loves). Curiously, readers familiar with Poe recognize the concept of giving one insult as being reasoning for murder (as taken from the later "The Cask of Amontillado"). Here, readers most likely question the murder of the old man based solely upon the narrator's dislike of the eye alone.
The main character killed the old man because of his "vulture Eye".