No (at least, not on my A6000, where the body is only 1" thick) - I would estimate the mount is ~5mm deep, and the shutter is 2-3mm in from that (I don't have the option to close the shutter when powered down, so can't easily check).
With a modicum of care, it should be easy to avoid touching the shuttter - my point is that it's people who are careless when changing lenses who are both most likely to end up with dust on the sensor, AND most likely to accidentally have something touch the shutter while it's exposed.