A nonadiabatic rate expression for hydrogen tunneling reactions in the condensed phase is derived for a model system described by a modified spin-boson Hamiltonian with a tunneling matrix element exponentially dependent on the hydrogen donor-acceptor distance. In this model, the two-level system representing the localized hydrogen vibrational states is linearly coupled to the donor-acceptor vibrational mode and the harmonic bath. The Hamiltonian also includes bilinear coupling between the donor-acceptor mode and the bath oscillators. This coupling provides a mechanism for energy exchange between the two-level system and the bath through the donor-acceptor mode, thereby facilitating convergence of the time integral of the probability flux correlation function for the case of weak coupling between the two-level system and the bath. The dependence of the rate constant on the model parameters and the temperature is analyzed in various regimes. Anomalous behavior of the rate constant is observed in the weak solvation regime for model systems that lack an effective mechanism for energy exchange between the two-level system and the bath. This theoretical formulation is applicable to a wide range of chemical and biological processes, including neutral hydrogen transfer reactions with small solvent reorganization energies.