Climbers Point
In my opinion, the problem is less about the actual hangboarding, and more about beginners not knowing when to stop and let their fingers rest. Since new climbers aren't usually used to the strain that climbing puts on their fingers and tendons, it's very easy to overtrain and to get injured.
This is true just by climbing alone. Add to that the element of hangboarding and you are in a very high risk of injury!
If a new climber can listen to their body, and also follow a strict program that will minimize the potential for finger injuries, even including hangboarding in the program could be fine. But, this will take a lot of self discipline that a lot of climbers don't seem to have.
Generally, I think that if you do things correctly, you could be fine doing finger specific training during your first year of climbing.
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.