I'm powering an infrared led with a forward voltage of 1.2 volts and a forward current of 100mA from a AAA battery. The LED calculator said I should use a resistor of 3.3 ohms. Do I really need a resistor if the resistance I'm supposed to use is so low?
If you don't use a resistor, the LED will die, perhaps not instantly, but it won't take long. A fresh AAA has got an internal resistance of between 100 mOhm and 300 mOhm (that's not MOhm in case you wondered) so with 0.3V to spare (with a new cell it would actually be up to 0.45 for a short time), a rough estimate of the current flowing is 1A to 3A!
While the battery wears down, current will go down too and you will soon have no power to the LED. A cell starts at around 1.65V and is considered empty at 0.9V, but your LED won't do a thing below 1.2V, so you will not be able to use up the cell.
If you instead use two cells and a Constant Current circuit, you will have a controlled current for the entire life time of the battery.
Btw. Why do you want to just have an IRLED light?