Does rock climbing damage your feet?

It for sure can if you wear the wrong shoes! It would also likely be a combination of other things such as wearing climbing shoes that are too small and landing badly, which could be bad regardless of the shoes, but wearing shoes that are not good for you would most likely increase the risk of bad landings.

Even without combining shoes and other factors, wearing climbing shoes that are not right for you, such as shoes that are too small, can cause damage by itself if you wear them for too long and if they are really too small.

Being well educated about how well sized climbing shoes should feel, mindful of if you got it wrong, and disciplined enough to listen to your body should help prevent damage to your feet though.