What is the theme of "The Cask of Amontillado," and why?

The theme of "The Cask of Amontillado" is the execution of a perfect crime. The whole story is taken up with Montresor's description of how he did it, not why he did it. It might be called a "howdunit" rather than a "whodunit." Poe makes this clear at the outset because he glosses over Montresor's motive with a single sentence:

THE THOUSAND INJURIES of Fortunato I had borne as I best could, but when he ventured upon insult I vowed revenge.

That is all in the past. The story is concerned only with the present. This makes it easy to visualize, almost as if it were a motion picture. The other important sentence in the opening is:

At length I would be avenged; this was a point definitely settled—but the very definitiveness with which it was resolved precluded the idea of risk.

The motive and the decision to be avenged are both "settled." What remains is to carry out the murder. This is a very adroit way of handling the story. Poe only needs to focus on one problem, which is the logistical one of luring Fortunato to his palazzo and down into his catacombs. The reader is not distracted by other considerations. The story has immediacy. Something could go wrong. Fortunato could start asking questions or balking at being led so far. Montresor has enough problems in the immediate present without needing to deal with whatever must have happened in the past.

The story ends almost as soon as Montresor finishes building his stone wall. Fifty years have passed since he completed it, but he does not say anything about what he has been doing for an entire half-century. Poe whisks the story to a conclusion with just a few words:

Against the new masonry I re-erected the old rampart of bones. For the half of a century no mortal has disturbed them. In pace requiescat!

So the past is disposed of quickly and the future—what happened after he finished building the wall—is disposed of equally quickly. Poe focuses on the little time period between the men's encounter on the street and the completion of the "immolation." Virtually everything takes place in the "present," and virtually everything can be told in description and dialogue, as in a play or movie.

The theme, of course, is revenge, but the story is concerned only with the execution of that revenge. Poe hardly even touches on Montresor's feelings about Fortunato and says virtually nothing about their relationship. This is strongly suggestive of what Ernest Hemingway was to say about his "iceberg principle" many years later.

If a writer of prose knows enough of what he is writing about he may omit things that he knows and the reader, if the writer is writing truly enough, will have a feeling of those things as strongly as though the writer had stated them. The dignity of movement of an ice-berg is due to only one-eighth of it being above water. A writer who omits things because he does not know them only makes hollow places in his writing.

—Ernest Hemingway, Death in the Afternoon