I don't ever shave horses, but in some circles they do get body clipped for winter riding, to help the coat dry faster when they sweat. For winter they don't clip the entire body, just those sections that would get the sweatiest. Those are also horses that are kept in barns and blanketed.
I do keep a small section of mane clipped short behind the ears, a "bridle path" so the headstall lies close and no fiddly mane gets in the way when I bridle. Most folks I ride with don't do that, it's out of horse fashion now I suppose.
A mane can grow back pretty quickly, and I hate to cut hers, but under the top loose hairs it's just a matted mess.
Right now I'm getting her accustomed to the sound of the scissors opening and closing close to her and just rubbing them along her neck and on top of her mane. I know she'll be fine once I start, but I expect she will jump around at the first few clips so it's not going to be pretty exactly.