hmmmm- looks like the 2010 TE310 & TE250 have different manuals.... and of course, I can't read the wiring diagram of the 250 easily. But it looks close to the 310 so let me throw a few suggested places to look at.
[later edit: Actually, a quick & dirty sensor test may be to just unplug it- if the fan stops (cold bike, key on, engine off?) then the sensor is bad. <--- yeah, do that first, that'll tell you fast.]
1st, try pulling the orange wire w/ a green stripe off of the fan relay (this may involve a lot of hand-waving if it is bundled in a multi-wire connector, as is likely). If the fan continues to run, it's probably a bad relay. (option 1 test: unplug the relay and test the 2 main terminals: open circuit means it's probably good; option 2 test- else just plug in another relay and try that one). This tells you it's the relay that is bad.
If the relay passes the above test, we have more of a mystery: put your money on a bad temperature sensor (hopefully, it's not a bad ECU). It's denoted as a WTS (water temperature switch) or #33 "coolant temperature sensor" on the wiring diagram. I'm guessing it is supposed to be at infinite ohms (ie, open) when the water is cold; you may find it closed (50ohms or way less) at any temperature- meaning it's bad. Just buy it and cross your fingers.
Try these two test and let us know what you find. BTW, my TE310 has been running fine, with the fan removed, in 90° F temps; although that's close to my personal riding limit- even while wearing an evaporative vest.
good luck.