I think the claim is that the shroud of Turin is woven in a way that was not known until around one thousand AD. If that could be proven it would mean that the shroud MUST be a fake. There may have been varying forms of textiles two thousand years ago but the selection would have been extremely limited compared to today, it wouldn’t have been simply a matter of having money.
Weaving patterns may not be typical of an era, but someone had to do it first, and often advancements are lost. (Damascus steel, for instance, (not pattern welded or simply layered steel) has yet to be reproduced.)
While certain fabrics can be ruled out as being from the era (synthetics, for instance), proving a pre-industrial revolution weaving pattern did not exist (at all, anywhere) at a certain date would be pretty tough, and opens the person making the determination to accusations of circular reasoning. (You can't simply date a pattern because it allegedly did not exist until a certain date, therefore it has to be after a certain date.)
There has to be other supporting evidence which would indicate that the material comes from a certain date, and in the instance of the shroud, exposed to fire, Carbon dating will just not be enough.
In the end, it becomes a matter of belief, not one of proof, at least until someone can demonstrate how medieval persons created the images in such a way that it fits the other evidence.