Like the policeman in the end of " The Tell-Tale-Heart," these policeman are generic characters, without defining characteristics, other than the fact that they are policeman. Like the police in "Tell-Tale" they drive the action by showing up and investigating. Unlike the police in "Tell-Tale" who seem like they plan on hanging out with the murderer all day, these guys are about to leave the house when the narrator inadvertently reveals the location of his wife's dead body.
In this case, the policemen get us thinking about the theme of "Justice and Judgment" in the story. Here's one way to look at it. None of the narrator's behavior up to the point he murdered his wife was illegal during the time Poe was writing – even if we think of the cat as allegorically representing a slave or a child.
A man could beat his wife, slave, child, and animals and be completely within the law. It was legal to kill slaves and pets, so long as they belonged to the person doing the killing. The death of the man's wife was necessary before the narrator could be brought to justice under the law. In some ways the policemen represent the limits of legal justice. The police are limited in what they can do. Even if they were afraid the man would kill his wife, they can't do anything about it until he actually does the deed.
We also wonder who called the police in the first place to report that the woman is missing? Remember, the police, by the fourth day after the murder, were hot on the narrator's trail. Did the man report his wife missing, or did someone else? What do you think?
In either case, the policemen would be required to suspect the man. Yet, they seem sure he did it. Their final search on the fourth day is the second such search. The policemen don't give up until they have searched the cellar three or four times. If the narrator is anything in person like he is on paper, then we know why the police suspect him so heavily. He's a super-shady guy.
But, maybe they already had their eye on him, and were just waiting for him to commit a legal crime so they could bring him to justice. Maybe he's been doing other bad things around town, or maybe the whole public cat hanging and subsequent fire brought him under their radar. What role do you think the policeman play in this story? What, if anything, do they tell us about justice and judgment?