Does the barber even feel like a character that belongs in this story? Sure, his little domestic squabble with his wife over the nose they find in a breakfast roll intros the main narrative. And his slovenly grossness and general dirty unpleasantness seems like it's going to be contrasted in some important way to how careful Kovalev is about his appearance.
But nope. Nothing the barber does with the nose prepares us for the idea that the nose is going to suddenly grow to human size and wander about by itself. Even though Kovalev's worries about his new nose-less appearance, it doesn't really seem to matter that out there somewhere in the world of the story exists another guy who is generally dirty and unkempt.
Really, the only reason to set the barber in the first place seems to be to throw out that final joke at the end—how the only way the Kovalev changes after his whole ordeal is that he no longer lets the barber hold his nose while shaving him.
On the other hand… it is just as possible that the barber is the most crucial element that sets the plot into motion. After all, the narrator takes a lot of time to talk about how
whenever Collegiate Assessor Kovalev was being shaved, and said to him, according to custom: "Ivan Yakovlevitch, your hands do smell!" he would retort: "But why should they smell?" and, when the Collegiate Assessor had replied: "Really I do not know, brother, but in any case they do." (1.18)
Maybe the nose escaped to get away from having to be so close to this guy's smelly hands every morning! Or, okay, maybe the nose escaped to teach Collegiate Assessor Kovalev a lesson about not being such a jerk.
Too bad it didn't work.