We present a model for the friction and effective mass of an oscillating superfluid He A-B interface due to orbital viscosity in the B-phase texture close to the interface. The model is applied to an experiment in which the A-B interface was stabilised in a magnetic field gradient at the transition field mT at 0 bar pressure and at a very low temperature mK. The interface was then oscillated by applying a small additional field at frequencies in the range 0.1-100 Hz. The response of the interface is governed by friction and by its effective mass. The measured dissipation does not fit theoretical predictions based either on the Andreev scattering of thermal quasiparticles or by pair-breaking from the moving interface. We describe a new mechanism based on the redistribution of thermal quasiparticle excitations in the B-phase texture engendered by the moving interface. This gives rise to friction via orbital viscosity and generates a significant effective mass of the interface. We have incorporated this mechanism into a simple preliminary model which provides reasonable agreement with the measured behaviour.