Well, I am neither a Catholic, nor an Italian so this hasn't got a lot to do with me although it's an interesting question..
This is between the Italians and the Catholic Church.
I suppose the Church can relocate if they don't like it - Set up a new tax exempt Holy See in the West Indies or something !

The Catholic Church owns property around the world actually - for example they have plenty of property in the UK.
Most of the time there is a monastery or a school or something on that property, but in some cases I think it's just pure investment. Not sure what their tax status is in other countries.

I spent a couple of days once at a monastery in connection with a school that a friend of mine attended for many years. We stayed in the guest quarters and I was pretty impressed - we stayed for free and were offered excellent food and good wine, also free. The monastery had an impressive art collection and the atmosphere was really nice - the monks were cool and very intellectual, not at all as I had expected. I definitely got the feeling that these were no "poor friars".