Yes you can, but you need something other then varnish to seal it. A good quality poly sealant is needed. Moisture in short order will cause mold issues on the cork if its not completely and properly sealed. This sealing is not as easy as you would think. So you really need to rethink the use of cork for the walls. If you miss one area or the wall is damaged and not re-sealed in short order, you risk mold fairly quickly which can make a nice wall look pretty ugly quite quick.
I would suggest that you contact the manufacture of the cork product your using and get their advice on what they suggest you use. There may be something on the cork product your using that may not react well with the sealants you choose.
Thanks so much for your response. If not cork, any other suggestions for covering up spearmint green wall tiles - the house is 50 plus years old. All tile in perfect condition - can't convince husband to tear it out. How can I get rid of that gawd-awful color on a shoe-string and DIY???
There are great epoxy paints that can be used. Our old home had Pink, Black and Blue tiles when we moved in. All in really good shape. We were on a new family budget so my wife sponged painted them. It lasted for over 10 years until we had the money to re-do the bath. You can cover them entirely or with the sponge method let a little green come through. If you have a few loose or damage tiles. Simply repair them with what ever you can get on sale to make the wall even and paint. Its a wonderful thing.
We'd like to add our expertise, but we agree with @Woodbridge Environmental Tiptophouse.com !