Despite being on the verge of committing a particularly nasty murder—walling up Fortunato alive inside his family's catacombs—Montresor is entirely without remorse. As far as he's concerned, he's doing the right thing; he's in the process of gaining revenge for the “thousand injuries” that Fortunato has alleged to have done him. Far from getting upset over killing Fortunato, he's really rather pleased with himself.
Even so, it's not very pleasant to spend all that time in the damp and dark of the catacombs. With its fetid air and lack of sunlight, it's not exactly conducive to good health. It's no wonder, then, that Montresor feels a little ill; as he tells us, he's feeling heartsick, an old-fashioned word that means “depressed” or “very sad.”
But as Montresor is quick to point out, his heart-sickness is entirely due to the dampness of the catacombs. That is what has made him feel quite down, not the killing of Fortunato. Just in case we didn't already know it, Montresor has no moral qualms about what he's doing. The only thing that makes him feel bad is not the unspeakably wicked murder he's about to commit but the dampness of the catacombs.
Even at this late stage, Montresor has the chance to change his mind and not go ahead with the murder. But he chooses not to and, in fact, goes about his grisly business with considerable haste, forcing the last stone into position before plastering it up.