Primordial magnetic fields (PMF) damp at scales smaller than the photon diffusion and free-streaming scale. This leads to heating of ordinary matter (electrons and baryons), which affects both the thermal and ionization history of our Universe. Here, we study the effect of heating due to ambipolar diffusion and decaying magnetic turbulence. We find that changes to the ionization history computed with recfast are significantly overestimated when compared with CosmoRec. The main physical reason for the difference is that the photoionization coefficient has to be evaluated using the radiation temperature rather than the matter temperature. A good agreement with CosmoRec is found after changing this aspect. Using Planck 2013 data and considering only the effect of PMF-induced heating, we find an upper limit on the rms magnetic field amplitude of B 0 ≲ 1.1 nG (95 per cent c.l.) for a stochastic background of PMF with a nearly scale-invariant power spectrum. We also discuss uncertainties related to the approximations for the heating rates and differences with respect to previous studies. Our results are important for the derivation of constraints on the PMF power spectrum obtained from measurements of the cosmic microwave background anisotropies with full-mission Planck data. They may also change some of the calculations of PMF-induced effects on the primordial chemistry and 21cm signals.