Not afraid, per se... but they do not want to destroy any more of the Shroud at this time. I think they are waiting for more maturity in the C-14 science before allowing it again. They were very dismayed at the sheer glee displayed by the so-called dispassionate scientists that announced the results from the 1988 tests... and the cavalier attitude they took to the sampling, discarding the agreed sampling protocols at the last minute (although I have to admit that the local See was complicit in that cluster f**k, as well).
Unfortunately, the 2002 "restoration" saturated the Shroud's reliquary with an pesticidal chemical that exudes a carbon pollutant that may contaminate the Shroud itself. There may be a saving grace to that "restoration:" the only source of uncontaminated test material are those carbonized linen fibers that were clipped away by that idiot Mechthild Flury-Lemberg, the so called textile expert, who thought the Shroud was still burning(!) from the 1532 fire, that were subsequently sealed in glass vials for safekeeping. Those are available and C-14 testing of them can do no further damage to the Shroud if they are destroyed in testing. All we can do is keep petitioning the Vatican for permission to test those clippings.
Sorry, I had not read this post before responding to annalex.
It has long seemed incredible to me how poorly the previous effort was done, with the resulting almost inexcusable fiasco. No one would ask for a repeat performance.
And in the Church's long perspective, a century is a mere moment in time. Still, it would be nice to learn in this life what will be immediately obvious in the next... ;-)