What time does it get dark in Italy in June?

There are slight variations depending on the exact location in Italy. For example in June sunset in Turin is later than in Bari simply due to the geographical location.
Rome is somewhat in the middle, and at the end of June sunset is at 8:50pm, therefore the civil twilight will continue for at least another half hour, while the nautical twilight will linger at least another hour after sunset. So in Rome there will be light at least until 9:30pm and will be totally dark after 10pm. In Florence it gets dark a few minutes later as it is located a bit more to the NW.
You can see the time of sunrise and sunset anywhere in the world in the website below or many similar:

http://www.timeanddate.com/sun/italy/rome?month=6&year=2015

Thanks so much! It is like putting together a puzzle to plan almost every minute of our days! :)

But, Mousie, thou art no thy lane [you aren't alone]
In proving foresight may be vain:
The best laid schemes o' mice an' men
Gang aft a-gley, [often go awry]
An' lea'e us nought but grief an' pain,
For promised joy.

.......Robert Burns

If you really want to use every minute, note the things you can do after dark as well.

Zoe said was I was thinking: your sightseeing shouldn't stop when the sun goes down. Many Italian cities and towns are wonderful after dark when some of the structures, fountains and piazzas are lit up, and walking old, narrow streets give you a sense of what life would have been like long, long ago.

Venice especially is awesome after dark...and check our Rick's night walks of Rome - I wasn't overly impressed with Campo Fiori, but the Pantheon is gorgeous at night, as is Trevi Ftn (well, our first visit it was...it's under restoration now) and the Colosseum.

In Venice, riding the vaparetto up and down the Grand Canal after dark is truly lovely. I'm sure a gondola ride would be even more so (although I didn't spring for one).