I'm not a cabinetmaker, but I've been around and seen a few built. Seems like they always veneer last, after the holes are already cut. the veneer is trimed along the edge with a sharp blade, such as a razor blade or utility knife.

I suppose there is no harm veneering the box before cutting the mounting holes as long as the cutting tools are very sharp and make a good clean cut. But all the cabinetmakers I've ever known finish the cabinet first, and then veneer, trimming the edges with a very sharp blade.