In a closed system, it is well known that the time-reversal symmetry can lead to Kramers degeneracy and protect nontrivial topological states such as the quantum spin Hall insulator. In this Letter, we address the issue of whether these effects are stable against coupling to the environment, provided that both the environment and the coupling to the environment also respect time-reversal symmetry. By employing a non-Hermitian Hamiltonian with the Langevin noise term and utilizing the non-Hermitian linear response theory, we show that the spectral functions for Kramers degenerate states can be split by dissipation, and the backscattering between counterpropagating edge states can be induced by dissipation. The latter leads to the absence of accurate quantization of conductance in the case of the quantum spin Hall effect. As an example, we demonstrate this concretely with the Kane-Mele model. Our study can also include interacting topological phases protected by time-reversal symmetry.