What perspective is "The Masque of the Red Death" told from?

This is an excellent question and substantially more difficult to answer than it seems!  The narrator does use the first-person pronoun "I" twice in the story, indicating that it is written from a first-person perspective; this would lead us to believe that the narrator is, on some level, a participant.  As the ebony clock strikes midnight for the last time, the narrator says, "And then the music ceased, as I have told; and the evolutions of the waltzers were quieted [...]."  Further, in describing the strange effect that seeing the Red Death has on the courtiers, he says that "In an assembly of phantasms such as I have painted, it may well be supposed that no ordinary appearance could have excited such sensation."  As a result of these two first-person references, we might assume that either the narrator is present at the masquerade (perhaps even the Red Death itself?) or that he is some kind of omniscient figure outside of the story (perhaps the author himself?). 

We know this narrator to be omniscient because he does know the thoughts and feelings of all the characters.  Of the courtiers, he says, "There are some who would have thought [Prince Prospero] mad. His followers felt that he was not."  Moreover, he describes them as eventually having "found leisure to become aware of the presence of a masked figure which had arrested the attention of no single individual before."  In order to describe their thoughts as well as their feelings and the persons of which they are or are not aware, the narrator must be omniscient.  Further, at the end of the story, the narrator describes the way in which the prince, "maddening with rage and the shame of his own momentary cowardice [in the face of the Red Death], rushed hurriedly through the six chambers [...]."  The narrator knows that the prince feels rage and shame and cowardice, knowledge that would not be possible were he not omniscient.

Therefore, we have an odd mixture of the first-person perspective with an attribute much more typical of third-person narrators: omniscience.  The narrator refers to himself as "I" but also has knowledge of all the characters' thoughts and feelings.  Thus, first-person omniscient seems to be the appropriate -- if atypical -- description of this story's perspective.