Thermal buckling deformation can significantly impact the operating performance of hydro-viscous drive. A thermal buckling finite element shell model was established with the nonlinear radial temperature as the thermal loading condition. The thermal buckling behavior of friction pairs was investigated under three different boundary constraints. Moreover, the influence of thickness and material parameters on the critical buckling temperature was discussed. The simulation results coincide with the failure modes of friction pairs in practice, and the most common ones are the coning mode and the potato chip mode. The ability to resist thermal buckling deformation can be improved as the thickness increases. In addition, the steel disc with outer edge simply supported is more prone to thermal buckling, because the critical temperature is minimum. The thermal expansion coefficient is the primary factor in thermal buckling study, which is inversely proportional to the critical temperature. These provide a theoretical basis for avoiding thermal failure of friction pairs in a hydro-viscous drive.