I think it really depends in which part of Tuscany you are going, expecially how high it is on the sea level.
Usually June in Tuscany is not too hot, the hottest months are july and august. On the hills above 400-500 metres on sea level you don't need air conditioning and there are not many mosquitoes/bugs. However, I think screens at the windows are always good, at least you can leave the windows open and get some air without a single mosquito getting in.
If you are staying in cities, or anyway not that high, at least a fan if not air conditioning would be good. On lower levels (below 400 meters) unfurtunately there are now african mosquitoes that have spread in all of Italy so screens might be good. But it really depends on where you are going: in some places you don't have many mosquitoes, in some other they can be really annoying.