We model the mass distribution of long gamma-ray burst (GRB) host galaxies given recent results suggesting that GRBs occur in low metallicity environments. By utilizing measurements of the redshift evolution of the mass-metallicity (MZ) relationship for galaxies, along with a sharp host metallicity cut-off suggested by Modjaz and collaborators, we estimate an upper limit on the stellar mass of a galaxy that can efficiently produce a GRB as a function of redshift. By employing consistent abundance indicators, we find that sub-solar metallicity cut-offs effectively limit GRBs to low stellar mass spirals and dwarf galaxies at low redshift. At higher redshifts, as the average metallicity of galaxies in the Universe falls, the mass range of galaxies capable of hosting a GRB broadens, with an upper bound approaching the mass of even the largest spiral galaxies. We compare these predicted limits to the growing number of published GRBhost masses and find that extremely low metallicity cut-offs of 0.1 to 0.5 Zo are effectively ruled out by a large number of intermediate mass galaxies at low redshift. A mass function that includes a smooth decrease in the efficiency of producing GRBs in galaxies of metallicity above 12+log(O/H)KK04 = 8.7 can, however, accommodate a majority of the measured host galaxy masses. We find that at z ~ 1, the peak in the observed GRB host mass distribution is inconsistent with the expected peak in the mass of galaxies harboring most of the star formation. This suggests that GRBs are metallicity biased tracers of star formation at low and intermediate redshifts, although our model predicts that this bias should disappear at higher redshifts due to the evolving metallicity content of the universe. Show Partial Abstract