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There have been claims that the repair portions sewn or woven (I dunno which) into the cloth after some fire damage centuries ago were the only bits used for the carbon dating. They weren’t. However, see the wiki-wacky-pedia:

http://en.wikipedia.org/wiki/Shroud_of_Turin#Radiocarbon_dating
http://en.wikipedia.org/wiki/Radiocarbon_14_dating_of_the_Shroud_of_Turin

the RC dating paper’s abstract:
http://www.nature.com/nature/journal/v337/n6208/abs/337611a0.html

If the carbonized portions of the cloth (iow, right at the point where the newer and older were joined) had been used, it *could* result in skewed results — but if so, the skew would be toward an *older* than actual date, rather than the reverse (iow, it would appear older, not younger than actual).

The samples were taken instead from the corners, which are image-free and basically just plain cloth, which minimizes damage and takes material from a part where no one is going to look anyway. :’)

I haven't got a subscription to Nature, so I can't read the abstract. However, repaired parts of the shroud weren't at issue. As I understood it, the theory works like this:

The cathedral was built with medieval timbers.
These timbers would contain carbon atoms reflecting the ratio of C₁₂to C₁₄ appropriate for the time when they were cut to build the cathedral.
R/C dating operates on the principle that organic material ingests atmospheric carbon and the amount of atmospheric carbon is fixed when the organic material dies.
The atmospheric percentage of C₁₂ to C₁₄ is known, so that if the sample precentage is different, we can apply the decay rate of C₁₄ to determine how many years have elapsed since the material stopped absorbing atmospheric carbon.
But, the fire would have released large amounts of CO₂ from the medieval timbers, which would contaminate the entire cloth, not just the parts that were repaired.
The R/C dating method involves releasing CO₂ from the material to obtain the carbon needed to study the ratio.
Because the fire contaminated the cloth with CO₂ from a later date, the test may not be correct.

This wasn't Bible-thumpers who came up with this. Even if the R/C method proved it to be a forgery, the Shroud of Turin still poses an intriguing mystery - how did they do it? We still don't know.