I am painting the entrance doors to all our rooms black. Do I need to paint the doors inside those rooms (closet doors, doors to en suite, utility closet) black also for continuity, or can I leave them white?
Rules aside, this will be fine; as long as this is the look your after. I assume all trim, door frame and the like will remain white. What this will do is accent all entry doors, which is cool.
There really isn't any "rules" that you need to worry about. The main rule is like the look your after.