Energy-frontier TeV colliders based on plasma accelerators are attracting much attention due to the recent achievements in multi-stage laser acceleration as well as the remarkable advances in electron- and proton-driven plasma accelerators. Such colliders may suffer a fundamental energy loss due to the radiation reaction (RR) effect, as the electrons lose energy through betatron radiation emission. Although the RR may not be critical for low-energy accelerators, it will exert limitations on TeV-class plasma-based colliders that need to be considered. In this paper, we have provided an extensive study of the RR effect in all pathways toward such colliders, including multi-stage plasma acceleration driven by the state-of-the-art lasers and the relativistic electron beam as well as the single-stage plasma acceleration with the energetic proton beams available at the CERN accelerator complex. A single-particle Landau–Lifschitz approach is used to consider the RR effect on an electron accelerating in the plasma blow-out regime. The model determines the boundaries where RR plays an energy limiting role on such colliders. The energy gain, the radiation loss, and the validity of the model are numerically explored.