Why does the clock have such a dramatic effect on the dancers in "The Masque of the Red Death"?

Time is an important theme in this story. The seven rooms symbolize the seven stages of life. Therefore, this one evening is construed in terms of a life span. As the guests move from room to room, and as the clock signals the passing hours, they move closer and closer to death.

The fact that the clock is "gigantic" illustrates how powerful and unstoppable time is. They cannot stop time and, therefore, they cannot prevent their own deaths. The clock is in the western room. Note the symbolism of the succession of rooms, from east to west, and the path of the sun rising in the east and setting in the west. This all emphasizes the passage of time. This allegorically parallels the life span and the notions of morning/birth and night/death.

When the clock signals the hour, the sound is: 

 . . . so peculiar a note and emphasis that, at each lapse of an hour, the musicians of the orchestra were constrained to pause, momentarily, in their performance, to harken to the sound; and thus the waltzers perforce ceased their evolutions; and there was a brief disconcert of the whole gay company . . . 

Each chiming of the hour interrupts the guests' good feelings. Each chime reminds them of time itself and that each hour, regardless of whether or not the plague might get to them, signals they are one step closer to their own deaths.

At midnight, the clock sounds twelve times. Once again, the music and the dancing stops while the clock sounds the hour. This gives the guests more time (twelve tones) to reflect upon the passage of time. This is the point. Each hour, there is one additional tone. So, each time the hour strikes and the music stops, the guests have increasingly more time to stop and ponder time, life, and death. Symbolically, it is at the twelfth hour that they have the time to notice the masked figure.