We compute the expected X-ray diffuse background and radiative feedback on the intergalactic medium (IGM) from X-ray binaries prior to and during the epoch of reionization. The cosmic evolution of compact binaries is followed using a population synthesis technique that treats separately neutron stars and black hole binaries in different spectral states and is calibrated to reproduce the observed X-ray properties of galaxies at z 4. Together with an updated empirical determination of the cosmic history of star formation, recent modeling of the stellar mass-metallicity relation, and a scheme for absorption by the IGM that accounts for the presence of ionized H ii bubbles during the epoch of reionization, our detailed calculations provide refined predictions of the X-ray volume emissivity and filtered radiation background from "normal" galaxies at z 6. Radiative transfer effects modulate the background spectrum, which shows a characteristic peak between 1 and 2 keV. Because of the energy dependence of photoabsorption, soft X-ray photons are produced by local sources, while more energetic radiation arrives unattenuated from larger cosmological volumes. While the filtering of X-ray radiation through the IGM slightly increases the mean excess energy per photoionization, it also weakens the radiation intensity below 1 keV, lowering the mean photoionization and heating rates. Numerical integration of the rate and energy equations shows that the contribution of X-ray binaries to the ionization of the bulk IGM is negligible, with the electron fraction never exceeding 1%. Direct He i photoionizations are the main source of IGM heating, and the temperature of the largely neutral medium in between H ii cavities increases above the temperature of the cosmic microwave background (CMB) only at z 10, when the volume filling factor of H ii bubbles is already 0.1. Therefore, in this scenario, it is only at relatively late epochs that neutral intergalactic hydrogen may be observable in 21 cm emission against the CMB.