Radio carbon dating is sometimes performed on relics; but it isn't typical because, well, they're relics - and radio carbon dating destroys part of it.
For example, radio-carbon dating was performed on the Shroud of Turin, believed to be the burial cloth of Jesus Christ, several decades ago. The testing done on the Shroud dated it to some time in the Middle Ages, which if correct would mean the cloth isn't the burial cloth of Jesus. But the results of the test are highly questionable and by no means conclusive, as Lagerbaer seems to believe.
The procedure was basically invalid because all of the samples were taken from the same section of the Shroud when they should have been from different parts. More importantly, that section of the Shroud had been repaired in modern times and the samples were very likely contaminated.
The Shroud has been damaged over the centuries by fires, etc. and has been repaired many times. Many, but not all of these repairs have been documented.
For more information about the carbon dating performed on the Shroud, see here:
http://www.shroud.com/pdfs/chronology.pdf
Also, http://www.shroud.com has a large collection of scientific articles about the Shroud, including high-resolution photos of it, for those who are interested.