Easy transistor circuit won't work

It looks to me that you are missing at least 2 things:

  1. A resistor between the Arduino output and the base of the TIP121, without this you are likely to burn out the Arduino pin; and

  2. A ground wire between the emitter of the TIP121 and the Arduino, unless the ground is connected via the power supply.

How do you know your IR LEDs are not working?