We have measured charge states deep within the ZnSe band gap and localized near the ZnSe/GaAs interface as a function of annealing temperature by means of photoluminescence spectroscopy, as well as cathodoluminescence from the cross section of a ZnSSe/GaAs heterostructure with a scanning electron microscope. Annealing produces new emissions at 1.9 eV and 2.25 eV which increase with annealing temperature and which vary with different growth conditions. Cathodoluminescence spectra show two similar features which originate within ZnSe and whose relative emission intensities vary with depth. Since the 2.25 eV feature appears only under Zn-rich growth conditions, these results indicate a role of both interface diffusion and crystal growth in the formation of this defect. We observe a proportional increase of the 1.9 eV and 0.9 eV peak intensities with annealing, indicating their complementary nature with respect to a common deep level. Finally, a detailed analysis of the relative intensities of ZnSe and GaAs features due to internal photoemission across the heterointerface suggests that the heterojunction band offset changes with formation of these new deep levels. All these results emphasize that multiple deep levels form near the buried interface with annealing which can dominate the heterojunction electronic properties.