The first step to refinishing the stained area of your dresser is sanding off the stain. If you were unable to remove it with a more traditional stain-removing method, you'll have to use sandpaper. Unfortunately, this process will also remove some of the dresser's paint, so it's important to be precise. Depending on the nature of the stain, you can likely use a fine-grit sandpaper to lightly grind away the stain.
After you've thoroughly sanded the stain off the dresser, wipe the excess wood dust off the area with a brush or microfiber cloth. Then, gently wash the area with soap and water to remove any other dust or grease stains from your fingers. It's important to have the affected area perfectly clean for when you begin to paint it.
Once you're ready to paint the dresser, make sure you have the exact shade of paint so that the new area doesn't stand out when you finish your work. Prime the area, if necessary, then add one or more coats of paint. If the paint is thick throughout the rest of the dresser, add multiple coats to give the dresser a smooth finish over the new area.
If the dresser has a glossy finish, cover your newly painted area with the same sort of finish. If you used a glossy paint, you may not have to do so. If the paint is matte, spraying a glossy clear-coat spray over the area will help give your newly painted area a clean, polished finish that matches the rest of the piece of furniture.