Montresor has no conscience throughout the events, but he may have one at the end.

Montresor is telling the story of how he killed Fortunato fifty years later, to a person he knows well.  We know this because he makes a comment at the beginning that the person knows him, and at the end he tells how much time has passed.  We do not know who he is talking to or why, but there seems to be indications of guilt.

Montresor may be confessing to someone, such as a priest or a friend.  He takes pains to explain why he had to kill Fortunato, and why he had to get away with it.

THE THOUSAND INJURIES of Fortunato I had borne as I best could, but when he ventured upon insult I vowed revenge. You, who so well know the nature of my soul, will not suppose, however, that I gave utterance to a threat. 

Montresor shows no conscience while he is actually luring Fortunato to his death.  He only pretends to care about his cold.  It is a ruse, to make Fortunato think that he doesn’t really care if he goes into the catacombs or not.

After bringing him into the kill room, Montresor does hesitate somewhat.  He seems surprised at Fortunato’s reaction, because the man screams and cries.  Montresor really only shows any emotion at the very end of the tale.

 My heart grew sick; it was the dampness of the catacombs that made it so.I hastened to make an end of my labour. I forced the last stone into its position; I plastered it up. Against the new masonry I re-erected the old rampart of bones. For the half of a century no mortal has disturbedthem. In pace requiescat!

The last line means “rest in peace.”  Between this and the twinge of conscience, and the fact that he is telling the story, we can infer that Montresor may be feeling some guilt.  However, it is also possible that he isn’t, and he is just a psychopath.

Yes, Montresor possesses a conscience throughout this story. It may not, however, appear that way on first glance, as walling up your enemy to slowly die of starvation does not seem like the act of a person with any sense of right and wrong. 

A chief clue that Montresor knew he was a committing an evil act, and felt bad about it comes near the end of the story. After he is almost done walling up Fortunato he notes that "my heart grew sick." This indicates that now that he has "won" and the euphoria is over, he realizes how terrible his act was--but that now it is too late to undo it. He tries to rationalize away the feeling by saying it must be from the dampness of the catacombs, but we know that is not true--the catacombs have been damp all along.

Further, Montresor is not simply acting as a sociopath, taking delight in walling up a random innocent person. His thinking may be warped, but he does believe Fortunato has wronged him: he does operate from within a moral universe. 

Another clue that Montresor's conscience is bothering him is that he is confessing the story 50 years after the fact, probably on his death bed. What has done has most likely weighed on his conscience all these decades.