I don't do Corian. It might be able to be sanded out and polished. I don't know the exact procedure, but I know you can sand it with an orbital and some automotive sandpaper and polish it with some stuff. I dunno, maybe just light it on fire.
Burns can cause deep discoloration. A blemish or even scratches don't require much more sanding than, say, a wood plank would. The trick is to be disciplined in using progressively finer grits, as there will be no concealing the swirl marks left by skipping stages. At the end, a good buffing with polishing compound will make it look like glass. Not hard, just tedious.