There's no evidence earthquakes cause neutron bursts.
The claims are based on "piezonuclear fission", a pet theory of the paper's lead author, Professor Alberto Carpinteri - that crushing solids (like rocks) can cause atoms to split and release neutrons. If this was true, you'd expect an event like an earthquake to release loads of neutrons. The only trouble is that that there's no reason to think it is true.
While Carpinteri's team claim to have observed high levels of neutrons after crushing rocks, "the experiments are badly described and no other groups have been able to reproduce them so far," nuclear engineer Ezio Puppin told Nature in 2012. The idea also contradicts a large amount of well-established nuclear science.
After all, nuclear fission generally requires the input of a huge amount of energy - slightly more than, say, hitting something with a hammer. Or even hitting it with an earthquake.
There's no reason to think that carbon dating is confused.
Carbon dating (which measures the number of carbon-14 isotopes in organic material, as they decay at a steady rate, making it possible to determine how old something is) isn't done in isolation. It's correlated with - and checked against - numerous other dating methods. If the carbon dating of items from Jerusalem around 33AD was inaccurate due to a radioactive earthquake, we might just have noticed that other carbon-dated artefacts from there didn't match the dates derived from other methods.
The idea that a burst of neutrons could have skewed the carbon dating results dates back to a letter in Nature in 1989, which suggested that the supernatural resurrection of Jesus could itself have caused a neutron burst that skewed the dating.
It's not entirely clear how neutrons would make the image of Jesus.
Neutron imaging is a thing - not dissimilar to X-ray imaging - but it usually involves a) a nuclear reactor to supply the large number of neutrons needed, and b) a specialised way of recording those neutrons, like x-ray film or digital equipment. A piece of cloth is not generally recognised as an especially useful medium to record neutron images on.
The man behind the claims isn't an expert.
The Daily Mail report says that the Politecnico di Torino, where Professor Carpinteri works, is "a well respected Italian University," which it may well be - but they neglect to say that it's an engineering school. It doesn't have an archeology department, a physics department, a geology department, or any speciality that would be relevant to the question of whether earthquakes cause neutron bursts that affect carbon dating and archeological artefacts. Alberto Carpinteri himself isn't a physicist, seismologist, or archaeologist - he's a structural engineer.
And it wouldn't be entirely true to say that Carpinteri is "well respected" - in 2012, over 1,000 Italian scientists petitioned Italy's research minister Francesco Profumo to block funding into Carpinteri's piezonuclear research on the grounds that "our international reputation could be damaged".
It makes literally no sense.
The paper itself is, uh... speculative. It doesn't really bother with pesky things like "making testable predictions" or "gathering experimental evidence". It doesn't explain why the shroud is literally the only bit of cloth in history that this happened to. It's very much from the "making stuff up because YOLO" branch of science.
Still, radioactive earthquake Jesus painting sounds cool.