No, actually it doesn't. The carbon stays the same as the source was still the original Flax and the soot from the fire is easily washed away. The explanation of the contamination that caused the spurious dating is now known: they tested a patched area.
>> The carbon stays the same as the source was still the original Flax and the soot from the fire is easily washed away. <<
Actually, if you put a cloth in a carbon-saturated atmospheric environment, it will even absorb and exchange CO2. So, no, you certainly cannot wash the soot away. Look at the shroud! The fire damage is very evident!
At the same time, you are correct about the patched area. Talk about a botch!