Will utilizing a TIP120 transistor to control the power from 2 9volt batteries in series powering a 12volt fan generate heat?
The transistor is being used in the diagram provided as a switching mechanism to turn the fan on and off. From the data sheet, the Arduino will output 5v which will then allow the circuit from the 9v battery to be completed through the tip120 and power the motor. This method is much more efficient and will create much less heat than replacing the transistor with a resistor. The tip120 is rated for 60V and up to 5A to flow from the collector to emitter. The heat you are noticing is probably from a low ohm resistor that isn't rated for the watts. So, for example if you have a 9v and a 5ohm resistor, then you will have 1.8 amps of load and require a 16+ watt rated resistor. (I = V*R)

Here is a link to the TIP120 datasheet: http://www.adafruit.com/datasheets/TIP120.pdf

Hope this helps.