Do Rome hotels require that you leave your room key at reception every time that you leave the hotel?
It depends on the hotel.
Most hotels everywhere these days use electronic key cards. There is always the occasional older hotel that still uses metal keys. They may ask you to leave the key at the desk, but it's never a city or country wide thing.
We were required to leave ours. I liked it as I didn’t have to worry about losing it.
The desk at our hotel in Rome was only open from 7am-10pm so it didn't have a policy like that.