You should check the wiring in the furnace. There are no standards for thermostat wire colors, so you can't always trust the color of the wires. Seeing how the wires connect to the furnace, is a sure fire way to figure out what's what.
The O (cool mode) and B (heat mode) terminals are usually used for a reversing valve, in heat pump systems. I'm not familiar with oil systems, so I can't say for sure what they would be used for.
Some thermostats have separate O and B terminals, like this.
Others have an O/B terminal, and use a jumper to select which should be used.
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.
Best Answer
So long as whatever you do de-energizes the wiring you're working on, you should be okay.
There's really just two concerns:
Shock hazard
You obviously don't want to get shocked/electrocuted, and only working on wires that are not energized is a good way to ensure this. This is less of a concern on the thermostat end, where you're only dealing with 24VAC, but there is 120V on the furnace control board.
Shorting out equipment
The other concern is when dealing with the wiring you can easily cause connections between stripped ends, which could do anything from simply turning the fan/furnace/AC on, to rapidly cycling equipment (which could damage it), to actually shorting something out -- such as the transformer itself -- and causing it to release the magic smoke, which renders it useless.