"My heart grew sick - on account of the dampness of the catacombs." How would you describle the narrator's reliability/objectivity? What does this say about the state of mind of the character?
Well, the narrator is, to put it quite simply, mentally unbalanced, and certainly not operating with a full grip on reality. Here he is, having planned this elaborate revenge, leading a costumed drunk guy down through the catacombs with the intention of chaining him to the rock and barricading him in with bricks, before leaving him to die, for some offense that has been yet unstated. So there's two ways to look at this: A) either his heart actually feels "sick" because deep down he has a conscience and he's really sickened by what he's planning to do, in which case his conscience apparently isn't quite strong enough because he does it anyway, or B) the narrator's actually feeling ill in the damp atmosphere of the catacombs, with no real thought about the impending murder, in which case he's even more of a cold-hearted killer than what might have been supposed. This narrator does not qualify as being reliable or objective in any sense of the word; creating characters with "a screw loose" was one technique Poe used effectively in his horror stories, as these type of people are not constrained by logic or reason, which adds to the frightening tone.