In this story, the narrator quite simply realizes that his dream had not been a dream at all.  He tells of how he “swooned” after receiving his death sentence from the Inquisition, and goes on to discuss the process of awakening from a swoon:  “In the return to life from the swoon there are two stages; first, that of the sense of mental or spiritual; secondly, that of the sense of physical, existence.”

This is a discussion of his attempts to understand what happened to him as he was semi-unconscious; he states that he can recall tall figures carrying him down, seemingly forever.  And, as he notes above that one first notices one’s mental or emotional state as one is roused from a dream, the narrator first feels “a vague horror at my heart.”  And only after this horror does he notice his physical environment – he is somewhere dark, dank, and flat.  At this point he is absorbed by “the mere consciousness of existence, without thought –” and then all at once his senses return to him, he is able to draw more details from his surroundings, and in this state of full awakening the details of his dream fall away.  As he says, “Arousing from the most profound of slumbers, we break the gossamer web of some dream. Yet in a second afterward…we remember not that we have dreamed.”

So, in the immediate shock of awakening in such a vile, ominous environment, the narrator forgets everything he has experienced after the trial, only able to remember it “vaguely,” after much intense mental labor.  And yet all this time he keeps his eyes closed, afraid of what exists beyond his eyelids, afraid that everything he had dreamed and everything he had felt was true.  And, unfortunately, in this fear he is correct.

In "The Pit and the Pendulum," what does the narrator dream?

I take it you are referring to the dream that the unnamed narrator experiences after being condemned to death at the beginning of the story. It is interesting to note that, before we are told what he actually dreamed about, Poe builds suspense by making readers wait to learn what happens next as the narrator gives us a meditation on dreams and consciousness. Finally, however, we are told that he remembers the following sequence of events after he regained consciousness:

These shadows of memory tell, indistinctly, of tall figures that lifted and bore me in silence down--down--still down--till a hideous dizziness oppressed me at the mere idea of the interminableness of the descent. They tell also of a vague horror at my heart, on account of that heart's unnatural stillness. Then comes a sense of sudden motionlessness throughout all things...

Note how the dream the narrator recalls vaguely could be interpreted symbolically as a descent into the grave, into hell or into despair as we picture silent figures bearing an inert person, whose heart is still, deep into the earth. This of course foreshadows the terror and despair that the narrator has yet to experience.