We present a general linear-response method for including nonequilibrium solvation effects (solvent friction effects) in variational transition state theory with multidimensional tunneling (VTST/MT) for calculating reaction rate constants in solution. The generalized Langevin approach is used to include a collective solvent coordinate into VTST/MT, and a general prescription is suggested for coupling this collective solvent coordinate to the solute, which is treated in its full dimensionality. The new formalism is illustrated by application to the aqueous free radical reaction H + CH3OH → H2 + CH2OH at 298 K. This reaction is treated with a linear mixing of Hartree-Fock theory and Austin Model 1 with specific reaction parameters (HF is parallel with AM1-SRP). The results with nonequilibrium solvation (NES) are compared to those obtained earlier with the separable equilibrium solvation (SES) and the equilibrium solvation path (ESP) approximations. We focus on the speedup due to solvation and on the kinetic isotope effects (KIEs). We calculate that nonequilibrium solvation decreases the rate constant by a factor of 2 but changes the KIEs by less than 2%. We also present results that show how the nonequilibrium effect depends on the solvation time and the strength of the solute-solvent coupling.