Do I need a health certificate to apply for a passport or visa?

Do not need a passport.

If it is handled at the entry-exit administration office, the time required is about 5 working days; If you go to the registered public security bureau, it will take about two weeks

Visa, short-term visa (such as tourist visa, visa to visit relatives and friends, etc.). You don't need a health certificate, as long as you have health insurance. But you need a long-term visa.