The effects of charge strength and potential range on the thermal diffusion factor and thermal conductivity of ionic systems was investigated using nonequilibrium molecular dynamics simulations. The potential model was a Lennard-Jones/spline function with an ionic tail of variable range and magnitude. The range was varied by selecting either a Coulomb or a Yukawa tail, and the magnitude was varied with the ionic charge strength. The study covers the range from a simple binary fluid mixture in the neighborhood of its critical point to a molten salt in the supercritical regime. Our main conclusions are that, under the conditions used here, the ionic charge inhibits the Soret effect in the Coulomb system (except when the valence is less than 0.02) but not in the Yukawa system. The thermal diffusion factor decreases from +1.03 +/- 0.13 for zero charge to -0.036 +/- 0.007 for univalent ions in the Coulomb case and to +0.098 +/- 0.004 in the Yukawa case. (The uncertainties are given as 1 standard deviation of the mean.) The thermal conductivity appears to be insensitive to the potential range for ionic systems. An analysis of the flux contributions reveals that the major contributions to the heat flux are the flux of kinetic energy and the collision energy flux. The potential energy flux contribution is not important, but it increases significantly as the system approaches its critical region.