How do you varnish a laminate kitchen worktop?
No. Laminate is meant to be impervious to liquids and oil finishes require the ability to soak into the surface.

Theoretically, you could apply a varnish (probably polyurethane)to the surface, but most laminate surfaces resist adhesion, so it may flake off on the sound sections and stick to the scarred areas. 

You probably could sand down the surface to give it tooth and better adhesion for poly, but I think you would be better off just doing a light sanding and polishing.  If that is not enough think abut a new surface. Maybe a new piece of laminate.

It all depends on what kind of surface you want and the intended uses.