How to remove burn marks on kitchen bench top?
As already stated... sanding is the answer.  Start with a course grit (small number, like 60) sand until the marks are gone.  It's much easier if you can borrow or rent a "random orbit sander", many can be hooked up to a vacuum to control dust.

Sand the entire area... ideally the entire countertop, so you don't create low spots.  Then work your way up (higher number grit) maybe 110, then 240 then 300 or so.  At that point you ought to be able to seal it with some kind of penetrating oil... mineral oil even works fine.  Look for "food safe" finishes at your local store.