How do you clean a burned glass coffee carafe?

Try adding a tablespoon of table salt and a half cup of hot water. This will create a course sludge that will (with a little elbow grease) power through the burnt carafe. If this fails, run some vinegar through your coffeemaker - a good preventative maintenance task for any coffeemaker - and let the carafe soak for a bit with the cycled vinegar. The acid should help break up the burn. Try adding a dryer sheet to some warm water and let sit over night.