The very first line of Poe's "The Black Cat" sets the stage for an ironic tale:
For the most wild yet most homely narrative which I am about to pen, I neither expect nor solicit belief.
Of course, the juxtaposition of wild with homely sets up a contrast between what is said and what is meant, and the declaration that the narrative does not want anyone to believe a tale he is taking the time and making the effort to tell, is also ironic. His further declaration that someone calmer, more logical, and less excitable that he will feel that the tale is
nothing more than an ordinary succession of very natural causes and effects
is nothing less than ironic, indeed, as he indicates his own insanity although he has stated in the third and forth sentences that he is not mad and that he does not dream--in other words, he is normal, too.
But, perhaps, the most startingly ironic situation in Poe's story is the scene in which the narrator describes how his wife accompanies him on some household errand into the cellar. When the cat gets in his way, nearly tripping him, the narrator becomes incensed and aims a blow with the axe in his hand at the cat. However, his wife intercedes for the cat. Simply because she gets in the way, he kills her instead:
Goaded by the interference into a rage more than demoniacal, I withdrew my arm from her grasp and buried the axe in her brain. She fell dead upon the spot, without a groan.
Clearly, this is no "homely tale" that the narrator "noted for docility" relates about his wife.
The link below is an excellent, detailed discussion of irony in this Poe short story, and I recommend you read it for a thorough examination of this topic. I'll simply highlight a few of the more obvious ironic elements.
First, the narrator's tone, as it is in so many of Poe's writings, is ironic. He's about to recount a horrific tale about his murdering his cat and his wife--yet he does so almost without care. "It's not that big of a deal, though some have made it a big deal" seems to be his tone. He treats the discussion of cruelty and murder as if he were retelling a memory from childhood which others might remember a different way. Just a misunderstanding, of sorts.
Second, it is ironic that he kills the cat he loves, and the cat he feels antagonism toward (Pluto) is the one which seals his fate.
Third, Pluto began as a cat who once loved his master, yet he was vilely mistreated. Once he loses an eye, he begins to "see" the narrator in a more accurate light.
Finally, the narrator is incredulous that a guy like him could ever become the man he's accused of being.
"From my infancy I was noted for the docility and humanity of my disposition. My tenderness of heart was even so conspicuous as to make me the jest of my companions."
As the events unfold, we find him to be anything but docile, humane, tender, or compassionate to man or beast.
Irony is a contrast between two things: what is said and what is meant (verbal irony); what is expected and what happens (irony of situation); and what the audience/readers know and what the characters know or believe (dramatic irony). Poe is the master of all three and he uses them masterfully in this story.