Is the reason it only happens with online bingo because this is the most intensive (CPU/GPU wise) tasks it carries out? - could be.
It is indeed normal for the fans to increase speed as the CPU temp raises - otherwise you would have the fan going full speed all the time.
A quick test for you - open calculator, switch to scientific mode, enter a large number (just press random numbers about 10 or so times) then press the "n!" key (under log key) it will appear to hang, then after a while it will tell you its taking a long time and should youcontinue or stop, continue a few times and see if that causes the fan to cick in as you describe with the bingo.