Remove any old finish from the surface by painting a stripper onto the wood using a wide bristle brush. Let the stripper sit according to the package instructions. The finish will look dull and soft when it's ready to remove. Scrape it off with a putty knife, being careful not to gouge the wood. Use a stripper designed for the type of finish that was previously on the dresser.
Clean off any residue by wetting fine steel wool with denatured alcohol and rubbing it on the surface. Replace the steel wool if it gets gummed up with the old finish. Continue working on the surface until no more finish is left. After it's dry, use a tack cloth to remove any dust.
Pour a de-waxed shellac onto a rag and rub it onto the wood to seal it. Rewet the rag as necessary to help the shellac go on evenly. Let the shellac dry overnight or as recommended by the manufacturer.
Apply a coat of lacquer to the dresser using a wide bristle brush and lacquer that is designed to be brushed on. Do not use any other type of lacquer because it will dry too quickly and leave streaks in the finish. Allow the lacquer to cure according to package instructions. Rub with fine 0000 steel wool to smooth surface and repeat until you have three to five coats of lacquer on the dresser. Use a tack cloth to remove dust between coats.