We present a hydrodynamic theory for systems of dipolar active Brownian particles which, in the regime of weak dipolar coupling, predicts the onset of motility-induced phase separation (MIPS), consistent with Brownian dynamics (BD) simulations. The hydrodynamic equations are derived by explicitly coarse-graining the microscopic Langevin dynamics, thus allowing for a mapping of the coarse-grained model and particle-resolved simulations. Performing BD simulations at fixed density, we find that dipolar interactions tend to hinder MIPS, as first reported in [Liao et al., Soft Matter, 2020, 16, 2208]. Here we demonstrate that the theoretical approach indeed captures the suppression of MIPS. Moreover, the analysis of the numerically obtained, angle-dependent correlation functions sheds light into the underlying microscopic mechanisms leading to the destabilization of the homogeneous phase.