Photomultiplier tubes, in particular multianode photomultiplier tubes, are often used in current accelerator-based nuclear and particle physics experiments for spatially resolved detection of single photons. With increasing luminosity and harsh radiation background, the radiation hardness of these sensors plays an increasingly important role. In an extensive experimental study we have investigated the effects of neutron irradiation, using a fission neutron source, and energetic 60Co γ-irradiation on commercially available Hamamatsu H12700/H8500 MAPMTs. In addition, individual components were irradiated, such as their glass windows and voltage dividers. Additionally, a p-terphenyl based wavelength shifting coating, which can be applied on the MAPMT window to enhance the photon yield, was also tested for its radiation tolerance.