Take your furniture to a well ventilated area and begin by peeling off any loose strips of varnish. Use only your fingers to peel off pieces of old finish, so as not to damage the hardwood underneath.
Put on a nose mask, and soak part of a oilcloth with paint solvent. Then apply the solvent to the surface of the furniture to dissolve the old layer of varnish. Rub the surface of your hardwood furniture in circular motions, with the grain, and against the grain to remove most of the old finish.
Take a break if the fumes become too thick. For a large piece of hardwood furniture, the process of applying solvent can take several days.
When you have removed enough of the finish, give the furniture a "clean rinse" by wiping it down with mineral spirits. Let the furniture dry completely.
Next, use a buffing pad to smooth and finish the hardwood surface. Buff the furniture with gentle swirling motions all over the surface. When you are finished, wipe the surface once more with mineral spirits.
Use a sponge brush to apply one layer of polyurethane (or other type of refinishing paint) to the hardwood furniture. Let the first coat dry overnight before applying the next, and store the furniture in a place that is free of dust and debris, yet still well ventilated.
Apply a second coat of finish, and let it cure for a few days before putting the furniture back in your house.