Electron tunneling through a dielectric barrier is considered with special attention given to questions relevant for STM experiments in dielectric liquids. The effect of the barrier dielectric response on the tunneling probability is studied using the effective Hamiltonian formalism for the polarization dynamics in the barrier, and two different theoretical approaches for the calculation of the tunneling probability: A generalization of the Bardeen's formalism to inelastic tunneling and the quasiclassical of Brink, Nemes, and Vautherin as expanded by Sumetskii. Although based on different approximations, both approaches yield similar results in the slow barrier limit, where their ranges of validity coincide. The approach based on the Bardeen's formalism relies on the adiabatic approximation and fails for fast barrier dynamics. The overall effect of the barrier dielectric response is to enhance the tunneling probability relative to the rigid barrier case. The enhancement factor is larger for thicker barrier, higher temperature and faster barrier dynamics. Both the elastic and inelastic components of the tunneling current show these trends in the relevant range of parameters.