You can probably (note the probably) get by without a fan on a track bike. I'd keep all the wiring intact just in case. A good strategy would be to wire a "warning light" in the same wiring that supplies the fan.
You wouldn't have to do anything else. The thermostat that controls coolant temperature in the engine is inside the small aluminum housing between the cylinders. Don't remove that thermostat.
The fan doesn't kick on until the engine coolant temperature (returning to the engine) goes above a certain setpoint.
Yes you can remove the fan. As long as you don't sit still too long you'll be fine. I actually ran my bike on the road for a season without a fan.... I don't reccomend it for the street but you get the idea.