Photoionization of HeH+ in the 40-80 eV photon energy range is investigated using the theoretical method of Sánchez and Martín (1997 Phys. Rev. Lett. 79 1654; 1998 Phys. Rev. A 57 1006), which includes all electronic and vibrational degrees of freedom as well as the interferences between all possible ionization and dissociation channels. Integral and differential cross sections in the energy of the remaining ions as well as energy positions and autoionization widths of the HeH+ doubly excited states of 1Σ+ and 1Π symmetries have been evaluated. The results show that dissociative ionization leading to He+(1s) + H+ + e- entirely dominates the photoionization spectrum and that resonant effects due to autoionization are barely visible either in the integral or the differential cross sections. This is at variance with photoionization of the isoelectronic H2 molecule.