It's generally up to you to insure the quilt if you're worried about damage. The shows will all "do their best" to keep the quilts safe, but they don't take full responsibility. Quilts do go missing sometimes.
I went to the Sisters quilt show earlier this month, one of my favorite shows. I love seeing the quilts outdoors and am glad that so many people, including some very famous quilters, are willing to display their quilts outdoors. There are people stationed around the show to watch out for people handling the quilts, and I'm sure they would notice if a quilt is taken down during the show. One year I was there when a thunderstorm suddenly rolled in. I helped take quilts down in the teachers' tent, and when we left we saw that quilts had been taken down all over town. They do get exposed to sunlight (depending on where they're hung) but little if any rain falls on them.