How big should the gap at the bottom of an interior door be?
There us also one other possibility. If you have HVAC and there is a grill in the room and there is no return in the same room, when the central heating/ cooling system is turned on it would create a high pressure in those rooms. Therefore sometimes a door cut higher is a viable solution to equalize the pressure. Removing the doors and cutting down the jambs IS NOT a viable alternative because the tops of doors and windows should plain out same height. 
I personally would do some type of detail on the bottom if the door if it was not necessary by design to equalize pressure. Get creative and there is a tremendous opportunity to do something quite unique. Such as a motion activated KED light strip as one example.