We study motion of vortices in arrays of Josephson junctions at zero temperature where it is controlled by quantum tunneling from one plaquette to another. The tunneling process is characterized by a finite time and can be slow compared to the superconducting gap (so that $����>> 1$). The dissipation which accompanies this process arises from rare processes when a vortex excites a quasiparticle above the gap while tunneling through a single junction. We find that the dissipation is significant even in the case $����>> 1$, in particular it is not exponentially small in this parameter. We use the calculated energy dissipation for the single vortex jump to estimate the physical resistance of the whole array.

24 pages, LaTeX references added, to appear in PRB