We study macroscopic quantum tunneling of interfaces separating normal and superconducting regions in type-I superconductors. Mathematical model is developed, that describes dissipative quantum escape of a two-dimensional manifold from a planar potential well. It corresponds to, e.g., a current-driven quantum depinning of the interface from a grain boundary or from artificially manufactured pinning layer. Effective action is derived and instantons of the equations of motion are investigated. Crossover between thermal activation and quantum tunneling is studied and the crossover temperature is computed. Our results, together with recent observation of non-thermal low-temperature magnetic relaxation in lead, suggest possibility of a controlled measurement of quantum depinning of the interface in a type-I superconductor.