Removing Varnish from Flooring

If it's a wood floor you'll probably need to rent a floor sander. If it's a linoleum/vinyl floor try paint thinner or turpentine to remove. If it's laminate flooring you should contact a laminate floor manufacturing company for instructions.

I don't believe you can possibly remove just the varnish. They do have clear varnish or polyurethane which you could put over the paint if you really think you need it. Why do you want to varnish it? If it's floor paint it should hold up just fine. I have painted my floors after using the proper floor primer and then used floor paint. I wanted a gloss finish for my kitchen floor and I was able to find it in a latex which was very easy to use. Previously I could only find gloss in an oil base which I wanted to avoid. It would probably be a good idea to talk to someone from a paint store but on the other hand I would simply sand/scuff the varnish and the paint would probably take...but possibly you'd need to go over it again with primer first. It's worth the extra pain of priming. If you were looking for a gloss finish with the ease of latex paint check out Home Depot. They carry some. If you don't want a shiny finish regular floor paint works great and you really don't need to varnish it at all. I've had some in a satin finish which has lasted years and I occasionally just touch up a spot or two with the left over paint.

Good luck!
P.S. I've even stenciled some floors and not varnished over them and they held up and some of the stencils were done with regular paint!