Is there irony in "The Tell-Tale Heart" and if so, how does it work in this story?
You ask a very good question. There is, indeed, a use of irony in "The Tell-Tale Heart." The most common form of literary irony is situational irony, which takes place when something happens in the story that goes against what we (the reader) expect or what makes sense.
In "The Tell-Tale Heart," there are several examples of irony. It is ironic that the murder's attempt to conceal his crime is what actually causes his incarceration. The fact that the killer confesses his crime when the policemen seemed to be quite relaxed and had no intention of arresting him is ironic. The death of one man with what may be considered a flaw (his eye) at the hands of one with a much more severe flaw (madness) is ironic; the murderer's flaw was hidden, but was infinitely more serious than the murdered's, which was visible, but of no danger to anyone but himself.
Overall, there are many ironies within "The Tell-Tale Heart." The principle of the value of friendship can be interpreted as being addressed via irony, as can many other variables. Within the story, several core human values are touched upon in ways that may be seen as ironic. This use of irony is typical of Poe's works.
Irony, if you recall, is A trope that involves incongruity between what is expected and what occurs. Here is a sampling of the many ironies in the story:
The narrator is a nervous and insane man, yet he claims he is calm and very analytical and logical. And he does analytically show (and calmly at first) how he became insane, even though he is actually trying to show to us that he is not insane, and finishes his story in a state of agitation.
The narrator is obsessed with the old man's eye which he fears, but it is the sound of the old man's heart that does the narrator in. There is another irony here: is it the old man's heart he is hearing or is it the narrator's own anxious heart beating strongly from guilt that he is hearing and mistaking for the old man's heart? Recall that earlier he states that he has an over accute sense of hearing.
See Buranelli, Vincent. Edgar Allan Poe. Boston: Twayne Publishers, 1977, for more information (others have given you a great eNotes link).
I would also like to add that it is quite ironic that he is basically the old man's helper and friend, a confidante of sorts, yet he turns around and plans his murder. In addition, the narrator claims that "he loves the old man and meant him no harm", yet he carefully plans the man's murder!
I would say that there is a sense of irony due to the fact that the narrator, who is clearly insane, is trying to convince the reader (and maybe himself) how sane he really is. But in this determination to show himself to be sane, he never denies having killed the old man - he fully admits his "guilt" in having ended the man's life, but chalks it up to the man's eye having caused this to happen. Check the link below for more information on the themes in this great story!