I saw a scholar conference about the subject a number of years ago, and they demonstrated that if you irradiate a piece of cloth you can completely distort the C14 readings (obviously). We don't know what printed the face on the Shroud of Thurin, but it's not paint because it's far thinner than what humans could even think of applying during the Middle Ages, so the "fake painting" hypothesis was ruled out since then. Furthermore, I think there were fabric analysis done to confirm that it was definitely something from about the Roman era in terms of quality and material.

I'll try to find some sources if you are interested in the subject.