Recent measurements of reactor-produced antineutrino fluxes and energy spectra are inconsistent with models based on measured thermal fission beta spectra. In this paper, we examine the dependence of antineutrino production on fission neutron energy. In particular, the variation of fission product yields with neutron energy has been considered as a possible source of the discrepancies between antineutrino observations and models. In simulations of low-enriched and highly-enriched reactor core designs, we find a substantial fraction of fissions (from 5% to more than 40%) are caused by non-thermal neutrons. Using tabulated evaluations of nuclear fission and decay, we estimate the variation in antineutrino emission by the prominent fission parents U-235, Pu-239, and Pu-241 versus neutron energy. The differences in fission neutron energy are found to produce less than 1% variation in detected antineutrino rate per fission of U-235, Pu-239, and Pu-241. Corresponding variations in the antineutrino spectrum are found to be less than 10% below 7 MeV antineutrino energy, smaller than current model uncertainties. We conclude that insufficient modeling of fission neutron energy is unlikely to be the cause of the various reactor anomalies. Our results also suggest that comparisons of antineutrino measurements at low-enriched and highly-enriched reactors can safely neglect the differences in the distributions of their fission neutron energies.