Forget about transistor gain. It has nothing to do with saturating a transistor switch.
It is clearly stated in the TIP120 datasheet that for saturation a base:collector current ratio of 1:250 should be used.
For a collector current of 1100mA, you should use a base current of about 1100/250 = 4.4mA.
BE voltage at that current is about 1.5volt (saturation graph).
And yes, the TIP120 gets hot with 1.1A collector current.
Saturation voltage is about 0.7volt at 1.1A, so 0.7*1.1 = 0.77watt is dissipated.
It could easilly get >50C without a heatsink. That's why we all use mosfets.
Leo..