Alitalia, Italy’s flag carrier, has the policy in detail on their website:
“To protect the health of passengers, the use of overhead bins for the storage of hand baggage will no longer be allowed on all flights operated in Italy. Passengers will only be permitted to bring on board small baggage, that can be placed under their seat such as, for example, handbags, backpacks, laptop cases not exceeding 36x45x20 cm. We invite passengers to deliver their baggage to the airport at the Check-in / Drop-off counter, to be placed in the hold, free of charge.”
Other European airlines have also updated their policies in recent days. British Airways has a section of their website dedicated to flights to and from Italy with the requirements:
“You will need to complete a Health Declaration form at check-in, and check-in any large hand luggage. Overhead lockers will not be used on your flight, so if your hand luggage will not fit under the seat in front of you, it will need to be checked in. There will be no additional charge for this, as long as it does not exceed the standard hand baggage allowance.”