In heterostructures consisting of different transition-metal dichalcogenide monolayers, a staggered band alignment can occur, leading to rapid charge separation of optically generated electron-hole pairs into opposite monolayers. These spatially separated electron-hole pairs are Coulomb-coupled and form interlayer excitons. Here, we study these interlayer excitons in a heterostructure consisting of MoSe₂ and WSe₂ monolayers using photoluminescence spectroscopy. We observe a non-trivial temperature dependence of the linewidth and the peak energy of the interlayer exciton, including an unusually strong initial redshift of the transition with temperature, as well as a pronounced blueshift of the emission energy with increasing excitation power. By combining these observations with time-resolved photoluminescence measurements, we are able to explain the observed behavior as a combination of interlayer exciton diffusion and dipolar, repulsive exciton-exciton interaction.