Short answer, no. Long answer, it depends and you'd need to be careful.
The issue isn't copying the artwork per se, it's copying the Vatican's picture of the artwork. They will likely release the images for free, but retain copyright over the digitisation.
Would you be caught for stealing a plate from an ancient book and incorporating it somewhere? Dunno. The Holy See is part of the Berne convention so you're still bound by their claimed copyright. Are they really going to go after you? I find it rather unlikely unless you're peddling images of the Sistene Chapel.
On the other hand if you made a reproduction of the image, say you found a drawing and manually vectorised it, then I think you would be safe - as after all, it's the photo that's copyright, not the manuscript.
In principle, taking a photo of a Van Gogh and putting it on a Tshirt is perfectly legal. The bans on photography in galleries boil down to:
- Light sensitive works
- Copyright on newer exhibits
- Profit making for the gallery (as many galleries in the UK are free, this is how they make their money)
- House rules, you may not be violating image copyright, but you may be violating house rules which they could kick you out for
In the US they can not claim copyright for this. This is very clear. See: https://en.wikipedia.org/wiki/Bridgeman_Art_Library_v._Corel.... and https://en.wikipedia.org/wiki/National_Portrait_Gallery_and_...
On the other hand, I'm not sure that there's a team of scholar-lawyers scouring the web for infringement.
I don't know how they could claim copyright on the original works, but they may have a case if you reuse their digitized copy of the original work. Much like a photograph of a piece of art may fall under a different copyright (that of the photographer) vs. the art itself. Also, the Vatican operates under a different set of rules than the rest of Italy, so I don't know if Italian copyright laws apply the same there.
Think: Retouch and sell as a painting.