If the holes are in the glass I fill them with JB Weld,
If in the wood I fill with clear epoxy and then sand flush. Paint with acrylics to match the wood color and grain it with a darker color matching to the grain and although I usually sand and refinish the entire riser you could get by refinishing just the area. Key is to block sand completely flush. sometimes I will use a shaving of the same kind of wood if I have a matching scrap in stock and cut out around the hole so I do not have a round hole and put the shaving in the epoxy with the grain running the same way as the riser. This makes a hole truly disappear. But then the painting when done well does too.
God bless and good luck, steve