Fortunato, more than a little intoxicated and dressed as a jester, is led from a party by Montresor, who feels that he has been wronged by Fortunato in countless ways, into his family catacombs with the pretense of showing off a cask of precious Amontillado, a rare sherry. Fortunato eagerly follows only to be chained to a niche in the wall where he watches as Montresor builds a brick wall to enclose him inside where he will be left to die with no hope of survival or rescue. For a detailed summary, check the link below.
Your question might concern what may have happened to Fortunato after Montresor walled him up in the niche and left him there. Naturally Fortunato would have died, but some writers have assumed that he died immediately of suffocation while others have assumed that he died of starvation. I believe it was Poe's intention to have the reader believe that Montresor's revenge included a long, lingering death for his victim. There was undoubtedly plenty of water, since the text specifies that there is water dripping everywhere and that they are in fact under a river. So poor Fortunato could have quenched his thirst by licking water off the rock wall to which he was chained. Poe also specifies that there is some air down there.
We passed through a range of low arches, descended, passed on, and descending again, arrived at a deep crypt, in which the foulness of the air caused our flambeaux rather to glow than flame.
Montresor also makes repeated references to the abundance of nitre. This substance contains a large quantity of oxygen. So it would seem that Fortunato could get water and some air in his confinement. The rough stone wall constructed by Montresor may contain enough chinks to allow the passage of some of the foul air from the other side. Montresor does not say so, but he might have left a few air holes in his wall.
So it would seem that Fortunato died of starvation while standing up. Perhaps in time his skeleton would have slipped through the chain and crumpled to the ground in the rags of his jester's costume. Montresor would have wanted his victim to suffer a long, lingering death. At the end, Montresor receives no answers from Fortunato, but that doesn't necessarily mean the man is already dead. That is unlikely. He has probably fainted or even refusing to answer.
Fortunato succumbs to the flattery of Montresor and is deceived by him to accompany him into the deep recesses of his cellar where he has allegedly stored the cask of amontillado wine. Montresor tricks Fortunato under the pretext of seeking his expert opinion on a cask of wine supposed to be of the rare and costly amontillado variety which he has allegedly just received,
"My dear Fortunato, you are luckily met. How remarkably well you are looking to-day! But I have received a pipe of what passes for Amontillado, and I have my doubts."
Montresor cunningly leads Fortunato deep down into the vaults of his sprawling mansion. Finally they arrive at the place where the ancestors of Montresor have been buried:
"We came at length to the foot of the descent, and stood together on the damp ground of the catacombs of the Montresors."
Montresor leads him to the extreme end of the vaults until they reach a dead end. Fortunato stood there confused, but before he could realize what happened Montresor chained him to the wall and quickly built a wall in front of him and buried him alive:
"A moment more and I had fettered him to the granite. In its surface were two iron staples, distant from each other about two feet, horizontally. From one of these depended a short chain. from the other a padlock. Throwing the links about his waist, it was but the work of a few seconds to secure it. He was too much astounded to resist . Withdrawing the key I stepped back from the recess...........I soon uncovered a quantity of building stone and mortar. With these materials and with the aid of my trowel, I began vigorously to wall up the entrance of the niche."