I'm on treatment for patches at my beard with Minoxidil. Will the thin hair coming out be destroyed if I shave it?

The thin hairs that are growing with minoxidil not be affected if you shave it.  But it may take the same amount of time for you to see it grow.  It will likely not grow any thicker.

There is no evidence that I am aware of to suggest that shaving hairs would result in their deterioration. Many believe the opposite to be true.

Shaving does not affect hair follicle growth patterns. An individual can shave as much or as little as he wants without concern for this altering the hair growth.