FAN ON

When the FAN switch on the thermostat is switched to the ON position, the thermostat energizes the G terminal. This tells the furnace to turn the blower motor on. If the furnace is equipped with a multi-speed motor, the motor will typically be in HIGH speed when the G terminal is energized.

FAN AUTO

When the FAN switch on the thermostat is set to AUTO, the blower motor will only be turned on as needed.

Heating

When in heating mode, the furnace itself controls the blower. When the thermostat gets cold, it calls for heat by energizing the W terminal (If you're using a heat pump, the O and/or B terminals could be energized as well). When this happens, the furnace initiates the firing sequence. The furnace will try to fire up, and if successful, will turn the blower on after a timed delay (20-30 seconds typically). When in heating mode, the blower will typically run at a slower speed (LOW, MED LO).

Cooling

When in cooling mode, the furnace doesn't typically do much at all. When the thermostat is too hot, it usually energizes both the Y and G terminals (If you're using a heat pump, the O and/or B terminals could also be energized). The Y terminal tells the condensing unit (or heat pump) to turn on, while the G terminal tells the furnace to turn on the blower (again, at HIGH speed).

Risks of having FAN switch ON when in heating mode

There are two minor concerns with having the FAN switch ON when in heating mode, but both are more related to efficiency than anything else. First, remember that when in heating mode, the furnace has a delay before turning on the blower. This delay is important, as it allows the furnace to heat up before it starts blowing. If there's no delay, the furnace will start blowing cool air around, and could take longer to reach peak operating temperature.

The second concern is that when the FAN switch is ON, the blower is run at HIGH speed. Furnaces typically prefer a slower speed when in heating mode, to allow the air enough time to pick up heat. If the air moves through the furnace too quickly, it may not pick up enough heat. This could lead to cooler air temperature at the registers, and a potential overheating risk for the furnace.

Furnace heating elements; whether gas burners, electric elements, etc., typically don't vary their heat output. They're either on, or off. The furnace relies on the air moving through it to remove the heat it generates. If the heat is not removed, the furnace will overheat. There are safety devices built in to the furnace, that turn the heating element off if it overheats. However, these devices are there to handle exceptions, and are not to be used as temperature control devices.

tl;dr

Blowing around cold air, and overheating the furnace are the two risks I can think of.

Here's a very simple schematic of a thermostat, that might be helpful.