This was a hot topic for a long time, and when they did C14 tests, it was "conclusive" that it was from the middle ages. Bah.
Any good archaeologist will tell you that C14 testing is only as good as it's purity in sampling.
a lesson in C14 testing, by RP:
Essentially c14 (I believe the first radiological method, that others are based on this process) goes like this: C12 (stable element - Carbon 12) is absorbed in living materials. It is assumed that c14 (unstable Carbon with extra electrons) is absorbed at the same rate. The testing process is essentially a count of the c14 that is in the specimen's remains compared to the c12, and yields a difference that can be applied to a chronology.
The challenge is that as with all radiological dating methods, there are many assumptions. The first assumption is the absorption rate is equal. This has been proven incorrect and is not even some "constant" at not being equal. What has been done to help, is to use dendrochronology (counting tree-ring pairs) and doing test of the tree rings themselves using c14 testing and then "calibrating" the resulting number.
The c14 test is then "calibrated" to some time ago about 6000-8000 years (interesting date range) because that is the oldest trees available to calibrate the dendrochronoligy with the radiometric c/14 calculations. So admitting (which scientists fully acknowledge) that c14 test must be calibrated, also verifies that the method is flawed to begin with.
There are c14 test that supposedly date prior to 6000bc, but are incorrect (as most scientists Christians and Non will fully admit), and without a calibration factor (remember the tree rings?), there is no way to know "how much", since there are too many assumptions that were initially made and are known variables now.
Even with that understanding that it is flawed, further making it less accurate is that the calibration itself is based on dendrochronology. while the counting of tree ring-pairs is pretty accurate, the assumption missed, here, is that it is based on yet another assumption that the absorption rate for all living matter is the same as the wood that was used in the calibration tests (not to mention assumptions that the absorption rate is the same for different woods). (on the order of about a thousand years off/too young at a dated period of about 5000bc, and it gets worse, the further back they go)
and the next thing to consider is that these dates are expressed using statistical approximation and standard deviation! that means at a single standard deviation, as example a date has a 67% chance that it falls within 25 years plus or minus. This could be 50 years, of the date claimed. Using two standard deviations (the number of deviation used should be noted when looking closely at these kinds of dates); the claim is now 95% chance that it is "accurate", but with a plus or minus of 50 years - 100 year difference - of the date claimed. This is not "proof".
Side note - So if you hear or read about carbon dating on dinosaur fossils or other fossils, they are wrong. a fossil is a rock, it is not biological matter, it cannot be carbon dated.
Essentially it comes down to a lot of conjecture, guessing and wanting certain dates to appear for various reasons. is it all wrong? no, but I have learned to question the questions that are used to come up with the questions asked for the questioning of the conjectures based on assumption.
With that said and in regards to the shroud - Whether it is "authentic" or not, I will say that I lean toward that it may be. what I do know is that C14 tests should never have been allowed for verifying this artifact.
Not only was it in at least one fire (heavy source of stable and unstable carbon elements - duh), it has been obviously handled by people over and over again over the years (referred to as biological contamination). when on a site in the field and discovering an artifact that is desired to be C14 tested, it is vary carefully verified that no other biological contaminants are (obviously) present (tree-roots, as a common example that might grow to/through something and causing bio-contamination), then using special tools it is placed in a special container to be sent to the labs (double blind testing is also usually done).
oh, well, sorry for rambling.