When analyzing Montresor's attitude or motivation, readers must keep in mind that he is another one of Poe's unreliable narrators. So, Montresor's attitude is not necessarily what Poe considers moral or logical.
Revenge (or delivering "punishment") is the main theme of the story. Montresor states at the beginning of the story,
‘‘The thousand injuries of Fortunato I had borne as best I could; but when he ventured upon insult, I vowed revenge.’’
Readers never discover what those injuries are that Fortunato enacted upon the narrator, but Montresor does not portray that as important. This demonstrates that Montresor's attitude toward revenge is inarguable--he believes so strongly in his right to revenge that he does not need to provide specifics about his motive.
Moreover, Montresor believes that he is qualified not only to obtain revenge himself (and not wait on a higher power or give his enemy an opportunity for redemption), but he also demonstrates that he is entitled to choose the mode and extremity of punishment. Whatever Fortunato did to Montresor (if anything at all) obviously did not kill or physically harm the narrator, but he does not care about the punishment fitting the "crime"--he wants only to assert his power and sense of justice.
Montresor's attitude toward revenge is that it is perfectly just and right to seek revenge when it is deserved. He believes Fortunato deserves revenge due to the "injuries" and "insult" Fortunato has inflicted on Montresor. In regard to punishment, Montresor feels that if one is punished for the revenge one exacts on one's persecutor, then the revenge cannot be considered effective or complete as a result. He says, "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." Therefore, another requirement of revenge is that the person on whom revenge is being sought must understand that the person they've wronged is now responsible for whatever fate befalls them.
Revenge is not only a matter of personal honor for Montresor, but also a matter of family honor. Montresor's family motto translates to "You will not harm me with impunity." This means that if someone injures a Montresor, one should expect to receive some punishment for that injury. Revenge, then, is not only a way to restore Montresor's individual, personal honor, but is required in order to maintain his family's honor, too.