My heat pump is located in the utility room/basement at the bottom of the unit is a vent that we can open or close.
PROBLEM: We just painetd the utility area. When the air is on and that vent is opened the smells are over the house. Oil based paint!

Can I CLOSE that vent until the area dries. Will this cause DAMAGE to the heat pump. I do not know if this is just a vent or the cold air return???????????????

Unless it's your only return vent in the place, close it. It won't matter in the short run in the least.