Should I keep my closet doors shut when the air conditioner is running? Will this help keep cooling costs down?
Generally speaking, any non-climate-controlled spaces you can close off - including closets - has the potential to lower your home cooling costs. The reasoning is simple: Any added square footage forces air coming out of the HVAC system go a longer way to achieve the desired room temperature.
One exception might be if the closet is damp. In this case, it would be good to have dryer air flowing through it so that clothes and other fabrics stored in there don't get moldy. This could also be accomplished via louvered closet doors, which let some air circulate through the smaller space.