Looking at Geofabrik’s taginfo for Italy shows a very low usage of heritage and heritage:operator compared with other countries and areas: England has 10k examples (and also has a distinct key listed_status), Bayern has 12k examples, but Italy only has 39 most of which (322) are World Heritage Centres.
Are there other ways in which protected buildings and monuments are tagged? Is there an open source of protected monuments?
Are there other ways in which protected buildings and monuments are tagged?
I think monuments are mostly tagged as archaeological sites, more recent protected monuments and buildings are probably mostly not tagged.
That’s what I feared! Wikidata is also lacking information of this kind about 20th century buildings.
There are sources that are openly accessible, and I guess also open data can be found, although maybe not systematically and complete for all areas. For example a search here (by municipality) lists 6231 results in Rome: http://vincoliinrete.beniculturali.it/VincoliInRete/vir/bene/listabeni (looks as if it isn’t open data, at least there is a copyright in the footer, and it doesn’t seem it lets you batch download everything).
This is the open data portal for the Lazio region: https://dati.lazio.it/ each of the 20 Regions has its own portal, AFAIK.
Here is maybe what you are looking for, PTPR - Tav. B - Beni singoli (a,b) - Dataset - CKAN
but the csv has only 147 lines 
Looking at it, there are many files with slightly different categories so you would have to merge them I guess: PTPR - Tav. B - Rispetto punti rurali - Dataset - CKAN
PTPR - Tav. B - Punti rurali identitari - Dataset - CKAN
PTPR - Tav. B - Punti archeo tipizzati - Dataset - CKAN
etc.
There is a map from the ministry of cultural goods here: http://vincoliinrete.beniculturali.it/vir/vir/vir.html