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It's a little messy, but it's nothing more than take off the old and put on the new. You can sand it off or you can use a Chemical Stripper. The Chemical Stripper is your best bet for a veneer's re-finishing. Then, it's simply follow the manufacturer's instructions on the re-finishing side.
Usually - sand smooth, brush the finish on, lightly sand smooth again and apply more finish. There's lots of brands to choose from and you'll probably be happiest with a Stain and Polyurethane combination product, they work very well and are more durable.