Who is the narrator, can she or he read minds, and, more importantly, can we trust her or him?
Third Person (Limited Omniscient); First Person (Peripheral Narrator)
Third Person
Pretty straightforward narration here. We get a standard-sounding narrator who sticks closely to the head of the main character, Kovalev. He mostly doesn't clue us into what other characters are thinking, so we learn about them from how they talk to Kovalev and what he ends up thinking about them.
But there are also some weird limits to the narrator's knowledge. The first two sections of the story both end mid-sentence with a variation on this little bizarro statement: "Further events here become enshrouded in mist. What happened after that is unknown to all men" (1.31).
Which, huh? Usually the only thing that a third-person limited narrator doesn't know is what's happening in the heads of secondary characters. This thing with not knowing the plot? Well, that points to a weirder snag.
First Person
And yeah—here's the snag. In the third section of the story, the narrator is revealed to be (spoiler alert!) just a guy who has been picking this story up from newspaper articles or gossip or whatever? Either way, suddenly we get to meet him in all of his confused-and-angry-man glory:
how came Kovalev not to know that one ought not to advertise for a nose through a newspaper? Not that I say this because I consider newspaper charges for announcements excessive. No, that is nothing, and I do not belong to the number of the mean. I say it because such a proceeding would have been gauche, derogatory, not the thing. And how came the nose into the baked roll? And what of Ivan Yakovlevitch? Oh, I cannot understand these points—absolutely I cannot. And the strangest, most unintelligible fact of all is that authors actually can select such occurrences for their subject! I confess this too to pass my comprehension, to—— But no; I will say just that I do not understand it. In the first place, a course of the sort never benefits the country. And in the second place—in the second place, a course of the sort never benefits anything at all. I cannot divine the use of it. (3.23)
It's like he's been telling us this story all along, skipping over the mysterious parts he doesn't understand, but then suddenly and out of nowhere he just throws up his hands and is all, "ugh, this story is just too crazy for me. Forget all this nonsense!"