Lay the painting on a flat surface. Wipe the surface of the painting with a lint-free cloth to remove loose dirt and dust. Avoid scrubbing or applying pressure to the canvas.
Clean the surface dirt and grime with a neutralizer product. Neutralizers are available at art supply stores. Apply the neutralizer to a piece of cotton wool. Gently wipe the surface of the painting with the cotton to remove the surface dirt. Replace the cotton wool with a fresh clean piece if the cotton becomes saturated with dirt.
Pour a small amount of varnish remover into a container. Varnish removers are available at art supply stores or online at sites such as blhaymondsupply.com. Dip a cotton swab into the container. Work on a 2 inch area of the painting at a time to remove the yellowed varnish. Avoid scrubbing the surface of the painting. Use a rolling motion with the cotton swab to remove the varnish. Rotate the swab over the 2 inch area. Inspect the swab to make sure it only has dirt and varnish and not paint on it. Don't dip the dirty swab into the container. Replace the swab with a clean one. Continue the process until the yellowed varnish has been completely removed from the oil painting.