If there is irony in the story, how does it affect our understanding? Is the theme ambiguous because of the irony? Explain how the irony functions in the story.

There is a great deal of dramatic irony in this story. Dramatic irony is created when the audience, or reader, knows something that one or more characters do not. In this story, we, of course, know of Montresor's plot against Fortunato. We don't know exactly what Montresor is planning to do, but we do know that he is plotting to effect Fortunato's "immolation," or complete destruction. Therefore, we watch, with building tension, as Montresor ensnares his victim, manipulating Fortunato into walking right into a deadly trap.
Despite the fact that Montresor insists that he must "punish [his adversary] with impunity," that the revenge will be incomplete, even void, if he must endure some consequence of it, there is evidence to suggest that he hasn't avoided consequence: that guilt has remained with him since the murder. He says in the final lines that it has been one "half of a century" since he took his revenge, and so it seems likely that he is now an old man, perhaps even confessing his sins to a priest on his deathbed. In the second sentence of the story, he addresses his auditor, describing the listener as one which "so well know[s] the nature of [Montresor's] soul." This provides further evidence that he might be speaking to a priest, someone to whom he has confessed for some time, someone who would understand him better than most. Thus, if Montresor has actually been carrying this guilt around for fifty years, it would seem that he has not escaped punishment and that his conscience has actually punished him for it—if he feels the need to confess it, then it must be weighing on him. Therefore, there is another dramatic irony at work here: Montresor believes that he got away with murder and that his revenge was really effective, but if we infer that he has carried his guilt with him for a half century, then we know something he doesn't: that his revenge, by his definition, was not complete.
We might, then, understand the story's theme to be that one cannot harm another person, no matter how justified it may seem, without burdening one's conscience; or, we might understand the theme to be that revenge injures the avenger as much as the avenged. However, if you do not believe that Montresor feels guilty for his crime, then this would eliminate this particular irony and change the theme as well.