Put the chair on top of a drop cloth in your garage. Polyurethane is toxic and emits a strong odor and should be applied only in a well-ventilated or open area, so do this with the garage door open.
Tape off the wood with painter’s tape to protect the upholstery (if any) from the polyurethane. As an extra precaution, affix a piece of masking tape to the edge of the painter’s tape and attach old rags to create an even wider protective layer.
Tint the polyurethane with artist’s oil paint, if desired. Squirt 1 tsp. of the oil paint in an old bowl and stir in enough polyurethane until you achieve the color you desire. Test the tint on a piece of wood before you apply it to your chair and let it dry; it is likely to dry a shade darker than the color it appears in the bowl.
Load the brush with polyurethane and spread the poly thinly over the wooden surface of the chair, keeping the brush at a right angle.
Feather the polyurethane by holding the brush at right angles to the surface of the chair. This will help spread out the poly so that it is thin and unlikely to drip. Walk around the chair to make sure that you have covered all the wood.
Let the polyurethane dry thoroughly. Depending on the climate and humidity, this can take between six and eight hours.
Touch an inconspicuous area of the chair, such as the bottom of a leg, with your finger to make sure that it is dry. If it’s still sticky, give it more time to dry. If it is dry, apply another coat or two, allowing for dry time between coats.
Throw out your tossaway brush or clean your varnish brush with mineral spirits before enjoying the newly protected chair.