The temperature and magnetic field dependence of the magnetic relaxation rate has been investigated at low temperatures (1.8 < T < 10 K) on two Tl2Ba2Ca2Cu3O10+δ samples (epitaxial thin film and sintered pellet). The temperature dependence gives evidence of a crossover in the mechanism of vortex motion, from classical thermal activation to quantum tunneling as temperature decreases. The field dependence of the relaxation rate indicates a crossover in the dimensionality of vortices, from three-dimensional flux lines to two-dimensional pancake vortices as field increases. For the thin film, the temperature dependence of the rate has been fitted to the theoretically predicted expressions for finite-temperature enhancement of the quantum rate in different regimes of dissipation. In spite of the similarity of the fits, the estimate of the ratio of Hall to viscous drag terms in the equation of motion indicates that quantum tunneling in this system occurs in an intermediate dissipative regime, where both terms contribute to the motion of vortices.