His teeth are rotted. How can I encourage him to fix them without hurting his feelings?

If you feel close enough to him to do so, say something like "Having good/healthy teeth is so important for your overall health because bacteria and things that live in your teeth and mouth can get into your bloodstream. Have you thought about getting your teeth checked out?" If you don't, bring up the connection between teeth health and overall health in a general conversation and use yourself as an example: "I've heard your teeth can really affect your overall health, so I made a dentist appointment to get everything checked out."