Cover your work surface with newspaper. Brush a heavy coat of chemical stripper onto the dresser. Let the stripper sit on the wood for the amount of time recommended by the manufacturer.
Scrape the old finish off the dresser with the putty knife. Use steel wool to get into hard-to-reach areas. Clean the dresser of stripper by wiping it down with a wet cloth.
Sand the dresser with 120-grit sandpaper to roughen the wood surface and remove the last of the old finish. Sand it again with 220-grit sandpaper to smooth the surface of the wood.
Brush a coat of stain onto the dresser with a clean brush, and allow the stain to absorb into the wood. Wipe off excess stain with a clean cloth. Let the stain dry completely.
Apply a thin coat of polyurethane to the wood with a clean paintbrush. Allow the polyurethane to dry, and lightly sand the wood with 220-grit sandpaper. Apply a second thin coat of polyurethane and let it dry.