A study is made of the effects of impurity radiation on the evolution of a reversed-field pinch plasma by means of a one-dimensional MHD simulation code that includes both plasma transport and impurity effects, and follows the plasma through a series of equilibrium states. Results are presented for the ZT-S, ZT-40, and RFX experiments with a fixed set of parameters, typical or envisioned, and various concentrations of impurities. Radiation barriers are encountered, and limits are found on the acceptable level of low-Z impurities that may be present in these devices. It is shown that the temperatures observed in the ZT-S experiment are radiation-limited. A criterion for radiation barrier burn-through is derived in terms of appropriate non-dimensional parameters and calibrated by comparison with the numerical simulations for the case of oxygen.