Finishing birch wood a person can use any acrylic finish. Try to stay away from nitrocellulose lacquer or oil based finishes. These finishes tend to yellow with age and would look bad on birch. An acrylic finish would remain clear. This could be a simple as using a water based polyurethane however any finish would need to be sanded between coats and you would risk sanding some of your markers off if you didn't put a number of coats on before sanding. Then water based finishes will raise the grain of wood making it rough. It will work but would need more coats and more sanding between coats to make smooth. If a person has the means of spraying a finish you could seal the wood with a vinyl sealer and use a type of lacquer known as butyrate lacquer, sometimes called cab acrylic lacquer. It would be a lot easier to work with and in warm weather the finish dries in 15-20 minutes.