Could you please give a little information about what the coop looks like and what parts you are considering painting? That might help on some specifics. Any wood that touches the ground should be protected. I don’t know if you are talking about a big walk-in coop on the ground with a dirt floor or an elevated coop with a wooden floor. Treated wood does not need to be painted but you are talking about untreated wood.
If the coop is on the ground, the splash area around the outside needs to be protected. That’s where rain or run-off from the roof can splash up on wood. If you have a wooden floor you probably should paint it and the area around the inside where the litter covers to slow that from rotting. If the poop builds up enough to be damp or the litter gets damp your untreated floor and bottom of the walls will rot pretty quickly.
Is it harmful to my chickens? No, untreated plywood is not harmful to your chickens. Not sure if that is what you asked or not.
Chickens and birds in general have sensitive respiratory systems. Miners carry canaries into mines to check for bad air. If the canary dies, get out before you die. Paint fumes can be dangerous to chickens. You can paint inside or outside the coop, your choice, but be sure the paint has cured enough so the fumes are not bad before you put chickens in there. If you paint the inside, air it our well. You should be careful breathing those fumes too in an enclosed space.
You can use different types of paint or sealers. Some might be better than others in some applications but as long as they are applied right and allowed to cure they should work out.
Good luck!