In "The Masque of Red Death", what does Prospero's running through all of his rooms and confronting the unknown guest in the seventh room symbolise?

You have asked a very astute question. The answer involves
an interpretation of the other symbols in the story to understand what is represented by
both the intruder and then by Prospero's movement through the seven rooms. It is clear
that this is a richly symbolic tale, and therefore the meaning is closely linked to
Poe's use of symbolism.

Let us remember that there are
seven rooms, each of a different colour. Seven is a key number that suggests the cycle
of life and time passing. For example, we have seven days in a week, and then we have
the seven stages of man. Let us also remember that the colour of the seventh room,
black, is richly symbolic of death, and likewise we need to recall that it is in this
seventh room that the clock (which again symbolises time passing) is housed. Of course,
Prospero and his guests have locked themselves away in an attempt to stop the inevitable
- to halt the ravages of time of live for eternity. The intruder, who could be said to
symbolise death, shows that this is impossible. It is strongly symbolic that as Prospero
follows the intruder through the other rooms to the seventh, he is, unknowingly, walking
to his death, as he meets the intruder at the final stage of life and dies
there.