What is the heat source maybe the fan was made to come on to keep the heat down where it was needed and what the heck are you doing with duct tape anywhere in a ceiling sounds like a mess to me. The easiest is to get an electrician to see what is going on beofr eyou burn the place down

Patricia Colwell Consulting ..they said .. until recently covered temporarily with duct tape

It's an older fan. Hampton Bay, I think. The duct tape was put on by maintenance to cover the gap until I can get a ceiling medallion installed.

It does not seem to be anything electrical. The only time the fan moves is whenever the space heater is on and facing it. As soon as I turn the heater in another direction, it stops.

This is also a rather drafty, humid apartment, and just wondering if it could be the heat rising so fast and escaping that it spins the fan blades. It can get spinning almost the same as the powered low speed when the heater is turned in that direction.

I think you hit the nail right on the head. Hot air rises cool air sinks. Outside it's called wind lol.

Airflow from the HVAC registers, of course. Happens in the lobby at work (2-story height). The four ceiling fans when turned off rotate when the HVAC is blowing, some more than others depending on the individual locations in relation to the registers.