What does the first-person narrative technique add to the story?
Poe's use of the first-person narration is to allow the reader to question the reliability of the narrator. Through Montressor's eyes, we are told that Fortunado insulted him over a thousand times; however, as the reader, we are only able to find one insult. It is Montressor who plots to kill his friend by taking advantage of his mental state and continuously filling him with drink. By the story's end, the narrator whom we initially believed in has turned out to be a maniac; thus proving his mental instability and the inability of the reader to truly trust his account.
It diverts away and stops himself from interpreting or hinting any actions regarding the author's behavior and gives Montresor full control over the entire story without external influence, without telling him what to do or what to leave out, leaving out a well-balanced story but giving a one-sided affair, an objective narrator telling his side of the story, the terrible and frightening details he put inside without any apparent recognition of what he did- the terrible crime he committed. It would also make the story less mundane but more interesting for the readers to digest.
With the first-person narrative, the story becomes much more interesting. We hear directly from the main character himself as he describes exactly how, and why, he killed Fortunato.
Because it's in the first person, the author's voice does not influence the story in one way or another-- all we have to go on is Montressor's first-person account, in all it's terrifying and calm detail.