Don't you feel like the narrator really has it in for pretty much all the characters in this story? Like, everyone single person he ends up describing gets a good punch square in the jaw just for existing? That, friends, is called sarcasm. And this story? It's dripping with it.
A good example of this is the narrator's treatment of the barber Ivan Yakovlevitch. The narrator literally can't help mocking him every time he comes up in the story. First of all, as he sits down to eat, the narrator says that he "donned a jacket over his shirt for politeness's sake" (1.5).
Now, it's true, the 19th century was a time when people were a little more buttoned up. No sagging your jeans down to your ankles in the 1830s. But this guy is so clearly gross and dirty, so clearly totally not a gentleman, that this jacket thing is just making fun. Same when he goes to throw the nose out into the river, and the narrator calls him a "worthy citizen" (1.19). Um, yeah. He's not worthy of anything and he's for sure not such a great citizen—that description is clearly totally tongue in cheek.
And then sometimes, the narrator isn't even subtle about it. Check out lines like "like every decent Russian tradesman, Ivan Yakovlevitch was a terrible drunk" (1.18). Okay, sure, this calls Ivan out. But it also makes fun of every single other small-business guy. And not only that, but by using the word "decent," what he's saying is that they all see being drunks as totally necessary—even good! Now that is a body slam of an insult.