"The Cask of Amontillado" is narrated by a man whose last name is Montresor (he and Fortunato are known only by last names). We learn his name when he and Fortunato descend into the catacombs and he names his family. Further, Montresor is narrating events that took place in the past: at the end of the story, he says that it has been a "half of a century" since these events took place. Therefore, the point of view is first-person objective because the narrator is a participant in the story and because he is narrating events after they have transpired instead of while they are happening.
If the story were narrated from a different point of view, it would certainly change dramatically. If, for example, Fortunato narrated the story, it would have to be while it was taking place (since he dies at the end), and since Fortunato was extremely drunk and missed the clues that might have helped him to understand what Montresor was intending to do (i.e. Montresor hid his identity, was carrying a trowel, and said something about turning back before it was "too late), the story would lose all sense of suspense or foreshadowing. He would just bumble along, enjoying the wine, thinking very highly of himself, until he found that he was being walled in at the end.