One more physicist here, with some experience into radiocarbon dating, and other methods.
Besides all the philosophical/religious/free-thinking/[put here your preferred stuff] arguments, we are dealing with a piece of cloth that, during the Middle Age, has been exposed to public in a church, exposed to flames coming from candles, oil lanterns and such. These are devices that burn carbon-based substances, and produce plenty of volatile carbon either as CO2 or soot. May also remember the 1532 fire, that nearly burned it ? As a consequence, doing C14 dating on this manufact is very, very questionable due to the external contamination of carbon. You can involve first class laboratories using AMS like it has been done, but the very basic hypotesis upon which the C14 method stands is flawed here.
To my opinion, C14 analysis isn't the best method to be used here, and it can at least give an upper limit for its age, nothing more.
I would also like to remember that palynology analyses dated the cloth to the period and location where tradition places it, and to my experience these methods are damn accurate (I work with people who routinely do this kind of analysis on very aged samples).
Just my 2c.
