Dentist removed tooth but left the root. He just said it would fall out anyway. Can I get an implant with an open bite?

The dentist should have removed the root. Like Dr. Dorfman said, you are susceptible to infection and your root will not fall out on its own, except in very rare and serious situations. I would advise you to see a qualified dentist or oral surgeon right away regarding your tooth extraction and implant. 

Removal of he crown of the tooth can result in a situation that leads to infection.  Normally roots dont "fall out" on their own. Implants can be placed with an open bite, but every situation is different.  Please see a board certified oral and maxillofacial surgeon to evaluate your current situation