The neighbor plays a small but important role in the narrator's story. As noted in the old man's "Character Analysis," the neighbor shows us that the narrator and/or the old man are alienated from their community. The narrator expresses fear that "a neighbour" will hear the old man's heart beating, and, sure enough, one little scream and the neighbor gets out of his or her bed, goes down to the police station, and raises enough flags to get the cops out the old house extremely quickly.
The fact that the neighbor's complaint was taken seriously suggests that he or she doesn't do this everyday—if he or she did, the cops might have ignored the complaint. So they're not really much of a character, but they play a vital role in this short story.