Personally for me that steps over the line. If I don't want to ride my horse you can't make me pay someone to do it. Some horses are just unrideable (lack of training or health issues.) I certainly don't want anyone to ride my crazy mare. They could get hurt. If its not in the contract you can't enforce it. The only thing you can do is ask them to leave.
Just to clarify the situation, the BO wants someone who is already taking lessons to pay to ride these boarder's horses. This person is not a boarder, just someone who is taking lessons on a lesson horse.
I don't think the BO has any right to give lessons on horses that do not belong to him/her. I don't think it's the BO business how much the boarded horses at his/her barn get rode. I think that when you board a horse, the only thing you are doing is paying to keep your horse on another persons property. The horse and all rights to it still belong to the owner.
Technically I would think this isn't really the BO's concern... but if s/he knew a lessoner who would do great on the horse and saw that the horse needed more exercise, I don't think it would be terrible if the BO presented the idea to the owner. Doesn't mean you have to say yes! I have a few fabulous riders who can't afford their own horse and they exercise horses for me. It's a win win situation for us! But as Kevin said, could be bad if they aren't experienced enough.