Generally not. In fact, horses that are provided with a free use shelter will often still stand out in the rain. They don't seem to mind it at all.
However, some horses do seem to hate the rain. And most horses will seek shelter if they hear thunder - they're tall and don't want to be hit by lightning. Horses will also try to turn their back to the rain or wind and may get mad with a rider who won't let them do that.