A major irony is the fact that the narrator planned a perfect crime and then was apprehended because he gave himself away. He confessed to killing the old man and showed the investigating officers exactly where he had hidden the dismembered body.
Another irony is in the fact that three investigating officers only chat with the narrator and show no suspicion of him or even of any foul play, but their blandness and failure to show any signs of suspicion are what ultimately cause the narrator to give himself away. He cannot believe they do not hear the beating of the heart as it grows louder and louder. He thinks they are playing cat-and-mouse with him.
It is ironic that a homicidal maniac should be trying to pass himself off as a genial host to these three investigating officers. The reader can imagine how nervous the narrator must actually feel. He says at the very beginning
True!--nervous--very dreadfully nervous I had been and am, but why will you say that I am mad!
It is ironic that the narrator is striving to show that he is perfectly sane and rational when everything he is explaining and describing shows just the opposite--that he is totally insane.