Irony comes in several varieties--verbal, structural/situational, dramatic--and generally heightens the our engagement with the text because we generally enjoy knowing more about a particular character's situation than the character does.
In "The Cask of Amontillado" Poe uses all three types. For example, the use of dramatic irony not only allows us to know more about what is going on than a character (in this case, Fortunato), we also clearly understand that Fortunato is never going to enjoy the Amontillado because he is being lured to his death--a fate of which he is blissfully unaware. Part of the our enjoyment of this irony is that Fortunato is a prideful, pompous man who thinks he has total control over his condition, and we know that this pride is actually leading him to his death.
Poe employs verbal irony very cleverly when he and Fortunato exchange comments about Fortunato's cold:
'Enough,' he [Fortunato] said; 'the cough is a mere nothing; it will not kill me. I shall not die of a cough.'
'True--true,' I [Montresor] replied.
The humor (fairly dark humor) is, of course, that we and Montresor know that Fortunato is not going to die of a cough, and so Fortunato is completely correct, but he is only correct because his death will come from a very different situation, one that he cannot possibly suspect at this point in the story.
Another example of both situational and verbal irony occurs as the two men descend into the catacombs, and they reach a point at which they discuss Montresor's coat of arms and the Montresor family motto Nemo me impune lacessit, which is Latin for "No one harms me with impunity." Based on Montresor's comment at the the beginning of the story that Fortunato has insulted Montresor to such an extent that Montresor decides to kill him, we have to be both amused and slightly horrified that Montresor's family motto is actually being acted out and, again, Fortunato is completely unaware of his situation and the threat to implied by Montresor's family motto.
Poe's use of irony in "The Cask of Amontillado" provides us with a great deal of foreknowledge of the story's conclusion, and the irony instills the narrative with very dark humor, which helps to lighten what would otherwise be a very grim story.