I haven't heard of any being hurt. Cats are pretty hardy, although I think they could get hurt if a tree has sharp branches or sharp ornaments. I don't think that even sharp branches or ornaments would do serious damage, though, because the impact of the falling trees would not be very hard. The branches cushion the fall. Trees tip when a cat gets near the top, where their weight is too much for top branches to support. So when everything falls over with the cat, it's the lighter weight top that lands on the cat.
Also, a tree has to be light enough for a cat's weight to knock it over. Average adult cat weight is from 9 pounds to 12 pounds. If the cat is inside the tree, close to or on the upper trunk, they are shielded from most ornaments and from the full weight of the tree.
I've only had two cats that climbed a Christmas tree. Both were just little kittens at the time. They could hardly have knocked over a stick, let alone a tree.
My current cat, Ember, climbed my tree during her first Christmas. She was only 4 months old and very small as the runt of her litter. It was so cute that I wish I had taken a picture or video. The whole time that the tree was up, she only swatted a couple low hanging ornaments to the floor. Each time, I picked them up and put them back on the tree.
Cats, especially kittens, learn a lot from observation and imitation. So when I started taking the tree down, she was super enthusiastic, like, "Wow! Mom wants to play swat the ball, too." While I took off the lower ornaments, she climbed up the inside to push off the upper ornaments from inside the tree. "Here, mom. You get the low ones and I'll get the high ones." When Iooked up after the first one fell, she peeked out from the branches looking so proud of herself that I couldn't help laughing.
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