How can I remove a burn mark on finished wood without replacing the piece entirely?
Go to a hardware store and get some good wood putty, filler or epoxy that can be molded and sanded.  Now sand out the burnt part of the molding.  Apply the putty in where you have sanded and have it roughly conform to the shape of your wood work.  Then let it dry to the time as shown on the manufacturer directions.  Now sand to match the rest of the molding.  Stain or paint to match.  Probably the hardest part is to match the stain, if it is stained.