In "The Black Cat," is the suffering of the narrator related to superstition about the cat or personal psychological problems?
It depends on how you interpret the story, which is one of the interesting things about Poe's stories and poems: they can have different meanings for every reader. Personally, I tend to favor the interpretation that the narrator has a psychological problem; that does not mean that superstition is completely out of the question, mostly because superstition is in people's mind. So the narrator himself may believe that the whole thing is because of his superstition surrounding the black cat, but the reader understands that his superstition is part of the narrator's psychosis.
Let us not forget that this whole thing started because the narrator got intoxicated one evening and removed the eye of Pluto, the titular black cat. This had nothing to do with superstition and everything to do with the narrator's alcoholism, which often has psychological consequences.
However, Poe certainly writes the story with the superstition about black cats in mind. When the narrator sees the only wall to survive the fire displays the image of a cat, he comes up with a scientific reason for why the image might have appeared. But how did the fire start in the first place? There is also the white spot on the second black cat that slowly takes the shape of the gallows.
In the end, though, the narrator's desire to mutilate and hang Pluto, and then to kill his own wife out of anger, are clearly problems of a psychological nature, not a superstitious one.