I turn mine off when the lights are off. Thats just me though because of odor control. The fan is not going to increase any co2 levels or anything it will just circle that ambient air that has 300 to 500 ppm co2 in it. If you want a cheap and easy way to increase ppm co2 even in a tiny grow is do what i recommend and go with a few pending on size of room, 2 liter bottles. Fill the 2 liter bottles with water about half way to the top. Then poke a hole in the bottle cap. Add about a full cup of flour and about a half a cup of sugar maybe more. Shake the bottle and bam you have co2. The more you put in the bottle the longer it will last. If you do this and turn the fans off the room will slightly rise in ppm's even with just 2 lliters. Yes co2 will promote growth.
I wouldnt take it down as long as it does not add humidity to the room.