My hands and feet go numb and tingle during and after cycling for long periods (15+ miles). Is this dangerous?

It can be. Sounds like there is a pressure neuropathy going on. The reason it can be dangerous is because at some point if you persist the damage can become permenant. So you have to figure out what the problem is and solve it. See a neurologist to see if they can figure out which nerve and why.
Probably not. Probably you are compressing nerves in wrists and ankles while cycling. Less likely there might be a positional blood flow problem. As long as your symptoms resolve completely after a suitable rest period, most likely no long term problems. But if weakness or no resolution of numbness between rides occurs, you are getting into trouble.