My wife shuts off bedroom doors to keep the pets out. My question is will this keep heating and air from not circulating through out the house correctly? It seems our great room stays cold and the bedrooms are warm. Thanks.

Your air distribution system SHOULD be designed to condition the great room and bedrooms evenly, closed or open doors. Running your fan in the "ON" position at the termostat may help, but what is really necessary is proper supply and return air distribution based on the "heat loss" and "heat gain" of each room individually.