Place the dresser on a canvas or plastic sheet to protect your floor. Remove the hardware, attached mirror, if there is one, and knobs. Fill the holes with wood filler and allow the filler to dry. Put on disposable gloves and a disposable face mask. Sand the dresser with a palm sander or sandpaper until the surface is smooth. Carefully remove the dust from sanding.
Paint a thin layer of white primer, using a brush on the edges and corners and a foam roller on the flat surfaces. Allow the paint to dry for 24 hours. Once the paint is dry, sand again to make the surface of the paint smooth, and carefully remove all the dust from sanding.
Paint a thin layer of flat paint on the dresser and allow to dry as before. Sand lightly to keep the surface smooth, and remove the dust before painting the next layer of paint.
Paint a second layer of paint, either flat or semigloss, depending on the finish you want. Carefully smooth out the brush marks if you are using a semigloss paint. If you use a flat paint, sand lightly again once the paint is thoroughly dry.
Distress the finish by sanding along the edges and corners with sandpaper to remove some of the paint. Replace the hardware and knobs.