The distribution of ions in the heliosheath—the region between the heliospheric termination shock and the heliopause—is important for understanding remote observations of energetic neutral atoms (ENAs). The ion distributions were estimated previously based on hybrid simulations of the heating and evolution of solar wind and interstellar pickup ions across the solar wind termination shock, but these estimates only provide the distributions near the shock. In this work, we use self-consistent hybrid kinetic simulations to investigate the effects of turbulence on ion distributions in the heliosheath. The simulations are compared against Voyager observations, constraining the feasible amplitude and compressibility of turbulence. We find that the heating due to turbulent dissipation can lead to a significant increase in the temperature of thermal solar wind ions. Both turbulent velocity fluctuations and the heating of solar wind ions increase the charge-exchange source for ENAs at low energies (around 100 eV), where current ENA models underpredict observations by more than an order of magnitude. However, the effects of turbulence are likely not strong enough to fully explain these discrepancies.