According to enth's write-up, the carbon in the "bio-plastic coating" would have to contain twice as much carbon as the shroud itself to change the shroud's carbon dating date by 1300 years. However, according to my calculations, this is almost right: it coating would have to be about 1.83 times the mass of the shroud itself (or, more specifically, for every pound of carbon in the shroud, there would have to be 1.83 pounds of carbon in the coating). (NOTE: This is a lot more than the 0.14, or 14%, estimate that I originally gave; my original calculations were wrong).

Here is my calculation: carbon-14 has a half life of 5730 years. Assume that the amount of carbon-14 in the atmosphere is constant from century to century (which isn't exactly true), and that 1 pound of carbon has 1 unit of carbon-14 in it. The shroud was dated in 1988 to have an origin of 1350 A.D., a 638 year difference; given this amount of time to decay, there would be 0.9257 units of C-14 per pound of carbon in the shroud.

If the shroud had been the burial shroud of Jesus, who died around 30 A.D., there would have been 1958 years for the C-14 to decay, and there would be only 0.7899 units of C-14 per pound of carbon. So, we need to add enough pounds of fresh carbon that the units per pound of C-14 rises to 0.9257. As we add fresh carbon to the shroud to increase the amount of C-14, we also increase the total amount of carbon, and the units per pound measurement is the total amount of C-14 divided by the total amount of carbon. Since there's 1 unit of C-14 per pound of fresh carbon, we calculate how many pounds of fresh carbon we'd have to add to a single pound of shroud carbon to get the ratio right.

0.7899 + X            <----- Total units of C-14
---------- = 0.9257   <----- Desired ration
   1 + X              <----- Total pounds of carbon

Solving for X, we get 1.83, so there'd have to be 1.83 pounds of carbon in the coating for every pound of carbon in the shroud. In other words, the shroud would have to be completely covered in slime and dust, which would be rather obvious to the scientists.

Another theory I've seen is that the flax the linen was made from fixes carbon-14 carbon-dioxide into chemicals more readily than the normal kind of carbon. I find it extremely unlikely that there's any kind of plant that fixes more carbon-14 than any other kind of plant, and if flax, a fairly important plant, did this, I'm sure that the scientists who do carbon dating would be aware of this and adjust for it.

A final theory that could be proposed, though I'm not sure if it has been. There's some theories that the corpse of Christ released radiation of the radioactive type while in the shroud, which caused the image. If this did indeed happen, it could have created a large amount of C-14 in the shroud, thus throwing the carbon dating date some 1300 years into the future.