Poe uses dramatic irony—when the audience knows or understands something that some character(s) does not—in order to heighten tension as well as to help convey the theme that, often, people simply cannot accept responsibility for their wrongdoings. The narrator of the story, a murderer, says that his purpose is to write down, for all the world to see, "a series of mere household events." There is nothing "mere" and little "household" about the events that he relates! He tortures animals, eventually killing his once-beloved cat, and, then, in his attempt to kill the other cat, he murders his wife instead and then walls up her body in his basement.
To say that these are "mere household events" shows that the narrator either cannot or will not recognize his responsibility as well as how terribly egregious his actions are. We, of course, do (as does the author), and so this creates dramatic irony.
The story makes use of situational irony as well. If someone had just spent a long time torturing and eventually murdering their cat because he could no longer stand the sight of the animal, how much would we expect this person to turn around and get another, similar, cat? Not much, right? Even after the narrator has hanged his first cat, Pluto—as well as been punished with a (symbolically) hellish fire that destroyed all he owned—he still took home a second cat who looked quite similar. This seems to show that, a sick person will compulsively continue to perform violent acts until they are forced to stop (via incarceration or death, for example).
"The Black Cat" incorporates a number of literary devices, including an unreliable narrator, symbolism, and irony, to reinforce its theme that wicked people cannot feel remorse because they do not take responsibility for their actions.
The unreliable narrator in this story is one of its most ingenious techniques. From the first sentence of the story, we are told by the first-person narrator, "I neither expect nor solicit belief." Mad we would be indeed to give it to him then! Therefore the reader must carefully study the narrator's words and read between the lines to find the truth of the tale. The narrator's effusive descriptions of himself in the second paragraph certainly must be taken with a grain of salt; we can see already that before the narrator degenerated into a violent man, he was already incapable of being honest with himself. Throughout the story, every time the narrator has a chance to take full responsibility for what he has done, he either blames something else (alcohol, his "disease," the cat) or he can't quite bring himself to be fully remorseful. Hence we have statements such as, "I blush to confess it," where he doesn't blush at the deed itself, but only at having to confess it, and "I experienced a sentiment half of horror, half of remorse," where he admits he does not feel the full weight of his guilt. Even in the very last line of the narrator's "confession," he notes that it was the cat who "seduced me into murder." Poe expects the reader to see through the narrator's biased interpretation of events and to understand that this wicked man, despite his evil deeds, is still unwilling to fully take responsibility for what he did.
The symbols in the story also point to the man's guilt and the way he continues to explain it away. The bas relief of Pluto that appears after the fire is a symbol of the depravity of the narrator, and that it needs to come to light. However, the narrator is able to come up with an explanation that "readily accounted to my reason, if not altogether to my conscience" of how the relic was created. He is unable to feel complete remorse, indicating he does not accept full responsibility. The second black cat is a symbol, like the first, of the man's guilt and seared conscience. The cat has a blind eye reminiscent of Pluto, and it has a marking on its chest that begins to look more and more like gallows. The narrator despises the cat, showing he has not come to terms with his past sins.
Finally, the narrator seeks to kill the cat, and instead murders his wife. Typical of his inability to own his actions, he hides the body by walling it up in his cellar. But just when it seems that he will once and for all get rid of his conscience and all reminders of his depravity, the cat calls out from behind the brick wall. The irony is that, although the narrator tried to kill the cat, he doesn't succeed, and the cat ends up giving away the narrator's guilt and sending him to the gallows. This irony reinforces the theme that a wicked person never takes responsibility for his actions, thus requiring outside forces to come into play to hold the person accountable for his evil deeds.