Yeah....re-reading this thread has shown Jabba's position to be utterly untennable.
The C14 sample was taken under some of the most scrutenized and controlled conditions I've ever heard about, much less experienced. To put it into perspective, when I take thermoluminesence samples the procedure is 1) cut rim off coffee can 2) scamper down into trench 3) push can into wet sand in trench, hopefully before the sand falls on me 4) ship now-full can to lab. The only supervision I had was the two guys taking bets on whether I'd be burried or not. Bear in mind, these samples were for a construction project--a LOT of money was riding on these samples. Yet because I've studied geology and was working under the supervision of a state-certified geologist, I was allowed to just go in and take the sample. Contrast that with the sampling of the shroud: more than a dozen people observing, cameras roling, the samples were immediately sealed up (mine were quickly sealed, but I didn't take the lid into the trench with me!) and immediately given to the lab representatives (chains fo custody generally amount to a paper you sign off on). This is REDICULOUSLY intense. The fact that they used trace-chemistry analysis to confirm that they sampled actual shroud fibers only makes it more so.
The sticky-tape samples amount to threads and dust taken from off of the shroud, which is highly problematic. We didn't have good ways of sealing things like the shroud off from contamination (technically a neutral word, and I'm being technical here) until fairly recently; certainly in the Middle Ages, and both groups agree that the shroud existed in the Late Middle Ages at least, dust, lint, stray threads, etc. could easily have reached the shroud. A fire obviously did, and does anyone here really think that keeping the shroud free from dust was the main concern of the people pulling it (and only partially succesfully, I might add) out of the flames? So the idea that the sticky-tape material is even admissible is laughable. It could come from anywhere, from any time, and therefore should be entirely ignored.
As far as the decay of organic compounds goes, that's extremely problematic. I use the decay of collegine as a first-order approximation for the age of bones in the field--if they're not human (and I'm not NEAR brave enough to do this to human bones!) I'll burn a small fragment with a lighter. If it starts to stink it's too young to be a fossil; if not, it might be. However, that can only tell you that it MIGHT be a fossil; it's not even close to definitive. And that's a real problem, considering my error bars are plus or minus centuries. And even my method is only accurate because of where I'm doing field work--relatively recent therapod finds indicate that organic material may be preservable for tens of millions of years, or may contaminate fossils that are older. I've heard of other compounds being used for dating, and they're always fairly inaccurate.
Simply put, a piece of cloth in a Medieval church isn't exactly going to experience the ideal setting for using the decay of organic mollecules for precision dating, and the method itself isn't useful for that anyway.
To make it worse, I know from other readings (I'm in the SCA with a Teutonic Order personna, so I've actually done a little research on the Church of the Middle Ages) that fraudulent holy relics were rampent in the Middle Ages, and the 1300s were a pretty bad time for it. Saints don't have that many limbs, even if you accept that St. Benedict may refer to numerous people through time, and Jesus wasn't nailed up by hundreds of nails on a cross bigger than a house. Many people, particularly wealthy ones, had little mini-alters they carried around with them (triptics, I believe they're called), which frequently contained holy relics. This actually caused a bit of a WTFuffle in the SCA about a year and a half to two years ago--another Crusader personna made a triptic and set it up, with a catsup-stained nail in it. He had no intentions of insulting the religious people there; it's simply something his personna would do, so he did it (it's actually a valid reason in the SCA). Some of the Christians that saw it took it a bit...hard. They started objecting. Which, this being a group of history nerds, lead to a rather in-depth, long-winded discussion about the whole holy relic thing. The take-home point is that holy relics were popular enough, and people were uneducated enough about history and even basic geography (not stupid, mind you, it's just that they really didn't CARE outside of their own lands), that a lot of churches made good money off "holy relics" only slightly more sacred than chicken bones.
That being the case, any relic that's spent any time in a church in the Middle Ages should be immediately viewed with extreme skepticism.
In fact, even if we grant Jabba's premise and assume that the shroud does date to the first century AD, it doesn't prove that the shroud is Jesus' shroud. All it proves is that the cloth is old. While it's difficult, it's not impossible to obtain such ancient cloth, and if you had a really good relic-maker, and a really good reason to risk it, it's entirely possible (and fits with everything we know about the Church in the Middle Ages) that the shroud amounts to an old bolt of cloth brought back from some crusade and made to look like what everyone assumes Jesus' burrial shroud would look like (which is really odd, considering the shroud is different from the biblical description--the head-cloth that was lying apart from the rest of it is conspicuously missing from the modern shroud).
From top to bottom the line of reasoning leading to the conclusion that the shroud is Jesus' burrial cloth doesn't make sense. Even if this WAS Jesus' burrial shroud it may in fact be impossible to prove at this point, and with all of the data pointing towards "It's not" there's no rational reason to conclude that it's anything more than a weird historical artifact.