Do UV-C phone cleaning devices work? Can they damage iPhones?

As far as sterilizing, yes UV light is highly effective at destroying both RNA and DNA viruses as well as bacteria. Hospitals have used high intensity UV light for decades for things that cannot be autoclaved.

I don’t know how well any home use consumer UV lamps work as they are usually pretty low intensity, so may take much longer to sterilize than commercial grade devices.

UV cleaning lights may fade the OLED display used on iPhone X series phones.