what happens if a hospital can't account for a patient?

Answer 1

They would probably consider it a discharge "AMA" (against medical advice) or "unauthorized absence." The doctors and other staff would still have to document the treatment that took place up to the absence.

If it is a mental health admission and the person is held involuntarily, the hospital may be legally responsible if the individual has a suicide attempt or harms someone else.

Answer 2

Probably nothing. They are not jails and if a person chooses to walk out , then they can't stop them.