I don't think it's necessarily the case that the narrator is afraid of the dark in general, but Poe certainly conveys the fact that he is afraid of the dark under these particular circumstances. The narrator is afraid to open his eyes, not for fear of seeing what is around him, but because of a fear that he will not be able to see anything. This turns out to be the case, and the narrator finds himself floundering in the darkness of "eternal night." His reactions to this are physical ones which underline how truly afraid he is. He becomes unable to breathe. It is the particular "intensity" of the darkness which he finds oppressive and physically stifling; either it causes the atmosphere to feel more "close" than it is, or it simply adds to the existing lack of air to make the situation unendurable for the narrator. He feels that he has been trapped in a dungeon, and is so afraid that he actually becomes unconscious at one point. He begins to compare the state of lying still in the dark to the state of being entombed, a quintessential human fear. The physical reactions to the darkness continue: the narrator is trembling and perspiring.
It is this continued reference to the physical effects of fear upon the narrator which especially help Poe convey to the reader how frightened he is of the darkness in which he finds himself. His fear is all-encompassing.