the narrator has to kill the old man, but the old man is completely unaware. There is also dramatic irony when the policemen come to the door in response to screams and are completely unaware that the old man is in the floorboards beneath them, but the audience knows, because the narrator has revealed it. 

There is situational irony when the officers come to the house, as well. The audience sees the narrator coming up with a plausible story about where the old man is, and about the screams that neighbors heard. Readers see him calmly chatting with the officers over tea. It seems like he's going to get away with murder. But then his own conscience betrays him as he imagines he can hear the old man's heart beating. He tells the truth about what he did. 

"Villains! Dissemble no more! I admit the deed! Tear up the planks! Here--here! It is the beating of his hideous heart!" 

In "The Tell-Tale Heart," we can find lots of examples of irony. Firstly, at the beginning of the story, the narrator tells us that he is not mad and that he is going to prove his sanity. What actually happens, however, is that the narrator tells us a story which proves the very opposite: he murders an innocent person, buries the body, but then hears a beating heart which leads him to reveal to his own guilt.

Secondly, there is irony in the narrator's feelings about the old man. On the one hand, he claims to love him but, on the other, he hates his "Evil eye" and plots to murder him.

Thirdly, we see the narrator go to great lengths to conceal the old man's body and, therefore, we expect that the narrator will get away with murder. What actually happens, however, is an example of situational irony because the narrator digs up the body and shouts his confession. This is ironic because it is the complete opposite of what the reader expected would happen.

In "The Tell-Tale Heart," Poe uses irony to help develop suspense.  Much of this irony is dramatic irony--situations in which characters are unaware of information to which the reader is privy.  Here are three examples of dramatic irony in the story.  First, the old man is totally unaware that the narrator harbors such ill feelings towards his "vulture" eye.  The narrator describes a "cold feeling" that runs through him every time the old man looks at him with the eye, so he resolves to kill him, unbeknownst to the old man.  Second, during the plotting of the murder, the narrator sneaks into the old man's room for eight nights to wait for the perfect moment to execute his plan.  But in the morning, the narrator acts warmly to the old man.  The old man still has no idea what will soon be his fate.  Finally, at the end of the story, the police officers are unaware that the old man has been buried under the floorboards, so they search the house and sit to have a drink with the narrator.  The reader is fully aware of the details of the story, while dramatic irony keeps important details hidden from certain characters in the service of developing suspense.

Dramatic irony is created when the audience knows something that one or more characters do not know, and it is often used to build tension in a text. Such dramatic irony is created as early as the first paragraph of Poe’s “The Tell-Tale Heart” when the narrator says,

I heard all things in the heaven and in the earth. I heard many things in hell. How, then, am I mad? Hearken! and observe how healthily—how calmly I can tell you the whole story.

This character, then, is telling us that he can hear everything going on in heaven and on the face of the earth, as well as a great many things that happen in hell. This is simply not possible. He believes that he is not mad, that his senses are simply incredibly sensitive and acute. He claims that he is both calm and healthy, and yet his insistence that he can physically hear most of what happens in heaven and on earth would seem to suggest otherwise. Further, his apparent defensiveness would also indicate that he is not at all calm. The character himself is, of course, wrong about his own abilities and the state of his mental health, though he does not realize his error. However, we do, and because we know this thing that he does not know, it creates dramatic irony.

Another example of dramatic irony in the text is created by the fact that we know that the main character plans to murder the old man with whom he lives, but the old man does not know this. This creates tension, too, as we await the murder.

Indirect characterization occurs when an author allows readers to form judgments about the qualities of a character rather than directly stating those qualities. The author can do this by showing readers what the character does or says so that readers can draw conclusions about the type of person he or she is.

In Edgar Allan Poe's "The Tell-Tale Heart," there is a good example of indirect characterization in the very first paragraph. The narrator remarks that whatever "disease" he has has "sharpened his senses" rather than dulling them. He claims that his sense of hearing has become much better. "I heard all things in the heaven and in the earth. I heard many things in hell," he says. In other words, the narrator actually thinks that he can hear, with his normal human ears, everything that is happening in heaven and on earth as well as many things happening in hell. There is little indication that the narrator is speaking figuratively, so it can be argued that the narrator is in some sense delusional or "mad," despite his insistence to the contrary. In this example, then, readers are not told directly that the narrator is mad, but readers are enabled to come to this conclusion as a result of the narrator's own speech and ideas.

The central theme of Edgar Allan Poe's short story The Tell-Tale Heart is ironic. The opening paragraph features the story's narrator vigorously objecting to any suggestion that his mental state is anything less than perfect:

"TRUE!—nervous—very, very dreadfully nervous I had been and am; but why will you say that I am mad? The disease had sharpened my senses—not destroyed—not dulled them. Above all was the sense of hearing acute. I heard all things in the heaven and in the earth. I heard many things in hell. How, then, am I mad? Hearken! and observe how healthily—how calmly I can tell you the whole story."

The irony lies in the story that follows, which inarguably depicts an individual driven over the edge by the recurring sight of the eye of the old man with whom he shares a home, an eye described as resembling "that of a vulture—a pale blue eye, with a film over it."  Which leads to the next example of irony in Poe's story: the plot by the narrator to kill the old man despite the latter's entirely inoffensive nature.  The narrator makes a point of emphasizing that he holds no ill will towards the old man -- "I loved the old man. He had never wronged me. He had never given me insult. For his gold I had no desire" -- yet such is his madness that he wants to kill the old man.  There is irony in his sentiments and in his actions, which do involve the old man's death.

A final bit of irony lies in the story's conclusion.  Having argued for his sanity, the narrator continues to display behavioral characteristics more like to the insane.  His guilt over his actions causes him to imagine that he is hearing the beating heart of the deceased who he buried under the floorboards, which causes him to cry out to the visiting police officers that he has killed his roommate.

Dramatic irony is definitely shown through the narrator’s remonstrations to the audience that he is not mad—he is not insane. The audience knows, even though the narrator has yet to accept it, that he is indeed mad. There is irony in the stealth with which the narrator creeps around waiting for the eye to be open, when it is the stealth that keeps the old man asleep and unaware. There is situational irony on the night when he startles the old man, opens the lantern after an hour or more and sees the eye looking at him. This is ironic because it is the open eye that he has been waiting for, and yet the eye fills him with rage. We would expect manic joy at the least that the deed could now be done. And there is great irony in that “his acute sense of hearing” caused him to hear the beating of a heart which could no longer—and did no longer—beat.

Enotes has some great resources:

http://www.enotes.com/tell-tale-heart

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.

Irony is one of the most commonly-used techniques in "The Tell-Tale Heart." Here are four examples from the text:

• In the first paragraph, the narrator declares that he is not mad but then proceeds to tell a story in which he clearly displays the symptoms of madness. That is, he kills an old man because he has an "evil eye."
• The narrator says that he was "never kinder" to the old man than in the week before he killed him. This is ironic because murder is probably the least kind act possible.
• The narrator mentions that the old man keeps the shutters down on his window to protect himself against "robbers." This is ironic because the old man does not realise that he needs protection from the man he lives with. 
• The narrator is proud that he was able to kill and dispose of the man "so cleverly" and "so cunningly," as a means of avoiding detection. In the final scene, however, the narrator not only confesses to the crime of murder but then shows the police the exact location of the body.