What would happen if one uses a higher power of eye glasses for hyperopia than they need?

Overcorrection in hyperopic patients is not well tolerated generally. These eyes are accustomed to maintaining an elevated degree of ciliary muscle tension just to neutralize the hyperopic blur at distance. They are furthermore conditioned to accommodate (increase the ciliary muscle tension) in response to blur in an effort to better focus. However, this actually makes the focus worse.

If the hyperopic patient is also developing presbyopia (a quite common situation), the overcorrection can help near vision, though. In any case, hyperopic overcorrection, however well-tolerated or not, will not harm the eye.