Mainly an individual comfort thing, but yeah, you can get sick from it just like if you rolled around naked in snow for a while or went swimming in really cold water on a cold day. Hypothermia. Fedor has some fat he carries around, so that might help insulate him a bit. If it is a hot enough day out that you don't feel cold, and it is socially acceptable where you are, go for it. I love doing it in the summer time back home, way out in the country. It feels like total freedom. Watch out for sunburns though.
It depends on if you feel that you are warm enough. Also, sometimes on a day that wasn't real hot, I would start out in a shirt and after running hard for a few miles, take it off when I get really hot. Or I would start running in the morning when it was coller and after while the sun was higher and the day had heated up enough so that I would take off my shirt and carry it or tuck it into my shorts. It is more how you feel and what is comfortable. You don't want to be cold or shivering when you are running.
Cool story: in cross country in high school, we had some really loose fitting enw jersies. I made the mistake of not wearing a t-shirt under mine on a meet day in November. The course was up on a hill, and it was a very hilly course. Because of this, the wind was blowing strongly the whole time, and it was cold. When I was near the end I noticed my chest was bleeding. Duringt he race, my nipples got so hard and dry they cracked and bled. The skin jsut broke right open. Ewwwww. Don't let that happen to you. I later learned that because of this, some runners put tape over their nipples, even when wearing shirts, to prevent this on cold days.