I went to Bella Italia a couple of years ago (loved it and want to go back) and I remember that many of the churches that my travel buddy and I visited had the dress code written on signs at the entrance. Before we left, we didn't even consider the possibility of dress codes, but luckily I had a safari jacket and my travel buddy had a cardigan. I understand that in Italy, it can be extremely hot, especially around this time of year, but I think that most people will be fine to put on a cardigan/light wrap at the entrance. I think that when I was there, with the exception of the Vatican, we spent an average of 20 minutes for each church. If I remember right, I think that the staff members were more focused on enforcing no photography when it was not permitted.