A friend recently tried to paint back-splash (above the counter tops) ceramic tiles to spruce them up. It didn't work worth a darn. The paint took over two weeks to dry and then it began peeling whenever she gently wiped the tiles with a damp cloth as in normal cleaning. She finally, secured some mineral spirits and removed the whole mess. I think you'd probably have the same results using varnish or any other finish. The tiles do not allow for absorption which is necessary for adhesion.
It is possible to have a professional come in and scarify (sanding done to roughen the tiles) and then finish with some sort of sealer. This, of course, destroys the ceramic hardness which is necessary for durability and stain resistance. You may be better off having a professional look at it and re-tiling over the existing ones.
Best of luck!