To settle this age old question of interior doors being open or closed, we must first address a topic called high room pressures.
Common signs of high room pressures are:
1. doors that slam shut by themselves whenever the air turns on.
2. a rush of air felt under a closed interior door when the air is on.
3. a resistance felt like someone is pushing back when you open an interior door and the air is on.
4. or an increase of allergic symptoms when at home.
A high room pressure can be caused by the placement and design of the ductwork, or uneven airflow across the ductwork from a lack of duct design or installation.
The problem with high room pressures only occurs when both:
1. The interior doors are closed. It is important to note that if your doors are never closed, high room pressures aren't an issue.
2. There is a high room pressure present. Listed above are the common signs of high room pressures that you can perform your own napkin check.
What happens when both your interior doors are closed and there is a high room pressure present is:
1. Every time the heating and cooling system kicks on, dust from the attic will get kicked up and enter the house.
2. You are essentially putting a big block on the duct running to that room. Think of blowing air into a bottle, you can't force more air into the bottle unless some can escape, the same thing is true with your rooms. The air has no way of escaping or circulating back out of the room and so no more than be pushed in.
3. The entire heating and cooling system will work harder because of the lack of free circulating air, forcing air through unnatural air leakage pathways connected to the attic and the outside.
Now some homes have jump ducts or transfer grilles already installed to help circulate air. A jump duct looks different than your regular supply registers. Usually, the registers dedicated for the jump duct are shaped differently and that is the only way you can tell you have "jumpers." The jump duct always comes in pairs of two, so for each jumper register in a room, there is a duct in the attic that attaches that register to a register in the hallway. So when you close your interior door, the air has a way to escape back to the return via the jump duct.
A transfer grille is the same idea only easier to install and identify. Transfer grilles commonly go above the door but can also be located in the interior door also and feature two registers, one on each side of the door with essentially a hole in the drywall in between them. When interior doors are closed, the air can escape through the transfer grilles (aka passive returns because they are passively bringing the air out of the room and back to the return).
So if you scrolled down to the end for the quick answer to your question, the answer is that interior doors should always be left open to circulate the supply air from the room back to the return. By shutting interior doors we are trapping the air in the room without a way to escape.