© 2015 The Authors. Published by Oxford University Press on behalf of the Royal Astronomical Society. We explore the galaxy formation physics governing the low-mass end of the HI mass function in the local Universe. Specifically, we predict the effects on the HI mass function of varying (i) the strength of photoionization feedback and the redshift of the end of the epoch of reionization, (ii) the cosmology, (iii) the supernovae feedback prescription and (iv) the efficiency of star formation. We find that the shape of the low-mass end of the HI mass function is most affected by the critical halo mass below which galaxy formation is suppressed by photoionization heating of the intergalactic medium. We model the redshift dependence of this critical dark matter halo mass by requiring a match to the low-mass end of the HI mass function. The best-fitting critical dark matter halo mass decreases as redshift increases in this model, corresponding to a circular velocity of ~50 km s-1 at z = 0, ~30 km s-1 at z ~ 1 and ~12 km s-1 at z = 6. We find that an evolving critical halo mass is required to explain both the shape and abundance of galaxies in the HI mass function below MHI ~ 108 h-2M⊙. The model makes specific predictions for the clustering strength of HI-selected galaxies with HI masses 106 and >107 h-2M⊙ and for the relation between the HI and stellar mass contents of galaxies which will be testable with upcoming surveys with the Square Kilometre Array and its pathfinders. We conclude that measurements of the HI mass function at z ≥ 0 will lead to an improvement in our understanding of the net effect of photoionization feedback on galaxy formation and evolution.