We present a detailed study of the non-Markovian two-state system dynamics for the regime of incoherent quantum tunneling. Using perturbation theory in the system tunneling amplitude \Delta, and in the limit of strong system-bath coupling, we determine the short time evolution of the reduced density matrix and thereby find a general equation of motion for the non-Markovian evolution at longer times. We relate the nonlocality in time due to the non-Markovian effects with the environment characteristic response time. In addition, we study the incoherent evolution of a system with a double-well potential, where each well consists of several quantized energy levels. We determine the crossover temperature to a regime where many energy levels in the wells participate in the tunneling process, and observe that the required temperature can be much smaller than the one associated with the system plasma frequency \omega_p. We also discuss experimental implications of our theoretical analysis.