Does zero chemical potential (particle number nonconservation) arise as a logical consequence of zero rest mass?

As for chemical potential, from the point of view of grand-canonical ensemble, it is a constant, coupled to the conserved number of particles: $$ mathcal{Z} = text{Tr} e^{-beta (H - mu N)} $$ The definition of chemical potential assumes the conservation of $N$. It seems not to be connected with the 'zero rest mass' property. For instance, one may consider the scalar or fermion field with the conserved $U(1)$ charge. The conservation of charge will also imply the conservation of number of particles, and one may have nonzero value for $mu$.

Another point, maybe worth mentioning, is that the thermodynamic equilibrium requires an interaction for particles to thermalize. Photons do no interact between each other, so they cannot achieve it.