Wood veneer is very delicate. Not every wood veneer furniture can be refinished. However if you have a ¼ inch of veneer with you, then you can easily refinish it.
Before refinishing the veneer, it is really important to assess the damage present on it. Any and all the damage present should be repaired at this time.
Once you are done with repairing the damage and your veneer is back into a perfect condition, you can now actually go for the refinish.
If the veneer is thin, you should remove the existing finish by rubbing lacquer thinner or denatured alcohol.
If the veneer is thick, then carefully sprinkle some sand over it. Take a tack cloth, and wipe away the dus with it. Do not, in any case, use water.
Take a foam brush and gently apply the new stain with it on the veneer. Wipe away any excess. Leave it to dry for a day. The next day, apply another coat and repeat until the desired color is achieved. Once you have applied the final coat on the veneer, leave it to dry for 48 hours.