Pull out all of the drawers from the dresser. Use a screw driver to remove the drawers' hardware, such as the knobs or pulls.
Put on dust and eye protection. Remove the old varnish or paint from the dresser with a sander. Be careful to keep the sander moving at all times so you do not generate any unintended mars on the surfaces of the dresser. Do the same with the drawers.
Run a damp cloth over the surfaces of the dresser and the drawers to remove dust from sanding. Let them dry before continuing.
Pour the finish of your choice into a paint tray. The finish can be tung oil if the dresser's wood has a nice grain, varnish polyurethane or stain. Apply the finish to the dresser and the drawers with a brush. Let the finish dry.
Put the drawer hardware back on the drawers.