The aim of this paper is to propose an asymptotic preserving numerical scheme for the three-temperature radiative transfer equations, which couple the radiative transfer equation with the electron and ion diffusive equations. The $H^T_N$ method is used for angular discretization, and the unified gas kinetic scheme (UGKS) is used for spatial discretization to obtain the asymptotic preserving property of the proposed scheme. Since it includes the electron-ion heat conduction, radiation-electron coupling and electron-ion coupling terms, the three-temperature model exhibits strong nonlinearity and strong coupling properties. By carefully dealing with these coupling terms implicitly and using an iteration method to solve the electron-ion diffusive system, we can adapt the $H^T_N$-based UGKS method for the grey radiative transfer equations to deal with this complex model. According to the asymptotic preserving property of the proposed scheme, it is not necessary in optically thick regimes to restrict the spatial step size smaller than the photon mean free path in order to approximate the solution of the three-temperature, two-temperature, and single-temperature diffusion limit equations under difference circumstances. The robustness and effectiveness of the current scheme are verified by several numerical experiments.