We investigate the superfluid–Mott-insulator transition of a two-dimensional photon gas in a dye-filled optical microcavity and in the presence of a periodic potential. We show that in the random-phase approximation the effects of the dye molecules, which generally lead to dissipation in the photonic system, can be captured by two dimensionless parameters that only depend on dye-specific properties. Within the mean-field approximation, we demonstrate that one of these parameters decreases the size of the Mott lobes in the phase diagram. By considering also Gaussian fluctuations, we show that the coupling with the dye molecules results in a finite lifetime of the quasiparticle and quasihole excitations in the Mott lobes. Moreover, we show that there are number fluctuations in the Mott lobes even at zero temperature and therefore that the true Mott-insulating state never exists if the interactions with the dye are included.