A perturbative renormalization group approach is employed to study the effect of a periodic potential on a system of one-dimensional bosons in a nonequilibrium steady state due to an initial interaction quench. The renormalization group flows are modified significantly from the well-known equilibrium Berezinski-Kosterlitz-Thouless form. They show several new features such as a generation of an effective temperature, generation of dissipation, as well as a change in the location of the quantum critical point separating the weak- and strong-coupling phases. Detailed results on the weak-coupling side of the phase diagram are presented, such as the renormalization of the parameters and the asymptotic behavior of the correlation functions. The physical origin of the generated temperature and friction is discussed.