Scrub the furniture with a rag dampened in oil soap to remove any film, dirt or oil. Rinse with a rag dampened in mineral spirits.
Sand the surface of the wood gently with 220-grit sandpaper to sand out any light scratches or knicks in the surface. Sand in the same direction as the grain. For larger pieces, use a random-orbit sander with 180- to 220-grit paper. Rub over the furniture with a tack cloth to remove any dust or other sanding residue from the wood. Again, work with the grain to prevent cross scratches. Rub the furniture with 0000 steel wool to complete the sanding. Wipe over it with the tack cloth a final time.
Select a Danish Oil finish in a color that matches the original. Compare the finish to sample chips available from your paint retailer to get as close a match as possible. Apply a coat of Danish Oil to the furniture with a clean, dry, soft paint brush. Work in the direction of the grain. Allow the first coat to dry according to the label instructions.
Rub the second coat in with 0000 steel wool. Soak the steel wool in Danish Oil and gently rub with the grain. This will allow the oil to penetrate below the surface of the wood and strengthen the grain of the wood.
Allow the second coat to dry. Wet-sand the surface gently to prepare for the third coat. Lightly brush mineral oil onto 220-grit sandpaper. Rub the sandpaper with the grain to smooth it. Sand the entire surface. Allow the oil to dry.
Brush on the third and subsequent coats, wet-sanding between each coat with 220-grit paper and mineral oil. Four to six coats is typical. Apply beeswax or paraffin wax to the surface for a higher-gloss sheen. Apply the wax with a soft clean rag and buff lightly.