If most of the veneer has chipped off, go ahead and remove the entire piece. You may need a heat gun to help loosen it. If only small amounts of veneer have chipped off, this is easily fixable. 

• Pull off any pieces of loose wood. If there are large loose sections, you can add in some wood glue and clamp down. 
• Apply wood filler generously. Let dry completely. 
• Sand wood filler smooth. Add more wood filler to any uneven spots. 
• Re-sand and wipe clean. 
• Paint the dresser and hide all signs of chipping and wood filler.
• Sometimes when a lot of wood filler is needed, it can be hard to get smooth. As a last resort, add a piece of wood trim. Cut the wood to size, glue and clamp. Then be sure to add wood filler around the edges. See my example here. 

Don't pass on those chipped veneer dressers. They can still be made beautiful. And don't forget to replace any missing or broken drawer slides.