The doors can be easily removed from the hinges to prevent any damage during construction.. With a proper finish on the jambs, the dust can be easily removed later.
That would keep them from getting dinged though. Draping a painters tarp or a blanket over the opening works well to control dust. If access is not needed, then plastic and tape to seal the entire opening.Or just lay a piece of 6 mil. plastic over the door and use blue tape.
By leaving the door on all the dust will not be blowing into the rest of the house.