It has previously been shown that a dye-filled microcavity can produce a Bose-Einstein condensate of photons. Thermalization of photons is possible via repeated absorption and reemission by the dye molecules. In this paper, we theoretically explore the behavior of the polarization of light in this system. We find that, in contrast to the near complete thermalization between different spatial modes of light, thermalization of polarization states is expected to generally be incomplete. We show that the polarization degree changes significantly from below to above threshold, and explain the dependence of polarization on all relevant material parameters.