Montresor, the narrator, seeks revenge on Fortunato because he feels that Fortunato has insulted him. He says, in the first line, "The thousand injuries of Fortunato I had borne as I best could, but when he ventured upon insult I vowed revenge." Thus, Montresor seems to feel that Fortunato has wounded his honor a great many times, and when Fortunato finally insults him, he can no longer stand by and take it any more.
Further, Montresor's family motto is "Nemo me impune lacessit," which means No one harms me with impunity, and his family arms show a human foot stepping on a snake which is, in turn, biting the foot. It seems that it isn't just Montresor's personal honor that Fortunato has wounded, but also Montresor's sense of family honor, since this family prides itself on avenging any crimes done to it. So, in order not to dishonor either himself or his family, Montresor seems to feel that he must exact revenge on Fortunato for whatever "injuries" and "insult[s]" Montresor has inflicted on him.