No. Mass doesn't "determine" the speed of time (a lovely phrase, BTW). It is the observer's motion through space that does that. However, for an observer to move through time at a speed faster than 0 it has to move through space at a speed slower than c, and to do that it needs to have mass. A massless particle can only transfer energy if it moves at the speed of light. Massless particles moving slower than the speed of light cannot transfer energy, and so cannot be detected experimentally, and so do not exist.