My heater fan will not shut off. Pulled thermostat off the wall and fan continues to run. Any answers are gratefully appreciated. So, update: We pulled the red wire off thermostat, and fan still keeps running. The only way to shut it off is at the furnace itself. But that turns whole system down. Now what?

The air handler is usually controlled by a thermostat (or simple timer) within the unit when in heat mode. The room thermostat calls for heat, the air handler gets warm (either by hot water loop, fuel burning, or electric) then when it is warm enough, the blower is switched on. This prevents cold air being pumped into the room. The blower stays on until the heating shuts off and the air handler cools down to below the preset limit (thus extracting the remaining heat in the unit).
My guess would be that the internal thermostat (or timer) in the air handler is stuck on.Unless you also have a humidistat connected to your air-handler and it is trying to add humidity to the room which some do must by blowing the air past a wet surface. If the water is off, the surface is clogged, the humistat is set too high, or you just have really dry air, then the unit doesn't know any different and it just keeps running without effect.

We do not have a humidstat connected. We have no problem getting heat. After the desired temp of thermostat is met the heat shuts off and the fan continues to run blowing air with no heat. When the temperature drops from the cool air blowing then the heat kicks back on again and warm air comes out of the vents.

For clarification, the thermostat I mentioned is internal to the air handler and not the one on your wall. It might be simple temperature sensitive switch mounted on the hot surface. It might just be stuck, give it a wrap.