No safe in apartment

Just leave cash and other valuables not needed that day in your case. Who will be going in that room? A cleaner, if that.
Technically you should carry your passport with you, and you may need to show it as ID.

People on this forum worry too much about passports. They are "everyday carry", not solid gold nor something every thief in Italy is after.

I keep excess cash, back-up ATM and credit cards, and sometimes my passport locked inside my suitcase everywhere I stay, which is usually in a budget hotel but occasionally is in a less formal accommodation like an apartment. I don't even make an effort to hide the stuff inside the bag, though it wouldn't be a bad idea to do so.

It is pretty risky for a hotel employee to steal from guests unless they don't care about their jobs. I keep my valuables in my locked suitcase. Yes, someone could pick it up and take it or cut it open. But, I believe that is way less likely than me forgetting it in a safe or when out and about losing them or having a pick pocketer swipe them. I know a couple people who have had passports pickpocketed (Budapest, Florence). They were fairly easily replaced. As far as other items, I have paper and digital copies of credit cards and any identification. For the most part, I don't travel with anything valuable other than electronics (ipod,ipad, iphone) money and credit cards

My parents never had a safe either in their house and never had an alarm system.
When I lived there and, later on, when my wife and I visited from the US, left the passports in the night stands inside the drawer, in plain view.
They lived in a large city (Florence) for almost 60 years, and their place was never burglarized.
Of course nobody can guarantee that your Lucca apartment will not be burglarized, but that is a rather unlikely event, and if thieves went inside, they would not be interested in your passports, but rather in your valuables, like cash, electronic devices, and maybe credit cards.
So keep your passports separate from your valuables, preferably in plain view, because if you put them inside some container with the rest of your valuables (like your suitcases), thieves in a hurry might take the whole suitcase with them. Remember that if burglars enter an apartment, a suitcase lock is not going to stop them from opening it, or worse, from taking the whole thing with them.
If you lose your passports, the US Consulate in Florence will replace it in minutes.
If you plan to leave passports at home while walking around Lucca, make sure you at least carry photocopies of your passport with you.

We typically rent apartments, and I also typically keep my passport locked up in my suitcase in the apartment. I have never needed the passport except when I am traveling on train, airplane, checking into a hotel.

I have actually never used a hotel safe. I think the only valuable item I bring is my passport.