I've heard this Charlemagne legend before but I'd love to see somebody put it to the test. I'm thinking the tablecloth would come out of the fire soiled with ash and bits of charcoal. I suppose it might still impress his guests but I don't think this party trick would work quite as well as the legend suggests.
Also, 'Salamander Cotton' is a delightful name for asbestos.
Less a tablecloth, more a placemat. And it is doubtful it was thrown onto the fire, where it would probably put the fire mostly out, rather than thrown into the fireplace to hang on something over the fire. There are also translation issues from royal to plain speak. Royals like this rarely did physical things. When the king is observed to "throw" something, it is more likely that he had it thrown by a servant who then placed the valuable object on a rack as planned. Readers of the time would not see a practical difference.
Legend has it that he threw a development team and a multitouch capacitive display in a fire and then pulled a fully built, finished iPhone from the flames.
In the early 20th century there were similar demonstrations done by asbestos manufacturers, and at least some of those were recorded on film. Fire resistance was basically the main selling point of asbestos, so they certainly marketed that heavily. I recall watching a video on YouTube of one long ago, but can't find it at the moment.
Also, 'Salamander Cotton' is a delightful name for asbestos.