What are four instances of irony in "The Tell-tale Heart"? 

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After the narrator has assured us many times that he is not insane, and he's described his repetition of movements and behaviors over seven nights as well as how he's lied to the old man to prevent him from getting suspicious.  Then he says that he "felt the extent of [his] own powers -- of [his] sagacity."  A person who is sagacious is one who makes good decisions and possesses sound judgment, two qualities the narrator clearly does not have.  This is arguably an example of dramatic irony, when the audience knows more than the character.

Further, the narrator's almost total lack of motive for killing the old man is ironic.  He says, "I loved the old man.  He had never wronged me.  He had never given me insults.  For his gold I had no desire."  Killing a person because one dislikes, even hates, that person's weird eyeball is not really a good reason to kill them.  A crime of passion can be understandable sometimes; we might pardon a murder if the murderer has somehow been terribly wronged by his victim.  However, the fact that the narrator really has no understandable reason to kill the old man is ironic (because our expectations of murder are different from its reality in this story).

After the murder, the narrator says, "If still you think me mad, you will think so no longer when I describe the wise precautions I took for the concealment of the body."  He dismembers the corpse in a tub so that no blood stains would betray him and he hides the pieces beneath the floorboards.  He believes that telling us this will make us think he is less insane when it will actually compel us to believe that he is more insane: this is ironic.

Finally, all along, this narrator has been telling us how reasonable he and his plan are.  We would never expect him to give himself away in the end, and this is exactly what he does.  Although the police seem to suspect nothing, he cries, "'Villains! [...] dissemble no more!  I admit the deed!'"  It is ironic that he confesses without provocation (except for his own terrible heart beat) or coercion because we would not expect most murderers to rat themselves out unless they were already suspected.