I'm planning on building an LED clock, but I want it to be lit only when I press the POWER button on my remote control. Using an IR receiver, which Arduino sleep mode should I use to reduce the current consumption but still be able to receive this interrupt?
I recommend reading ArduinoSleepCode. Where it demonstrates placing the avr into a sleep state and using an interrupt to awake from the sleep state. In your case you can connect the output of your IR demodulator to the interrupt pin, to simular awake.
Ken Shirriff's IR library allows you to use any pin as a receive input, so that you can use INT0 or 1 on D2 or D3. If you need to use a different pin, the Microtheron's github fork uses Pin Change interrupts. (I am not sure if stock sleep.h will awake from PCint, like int0 or 1 will, without tweek. Section 10.1 of data sheet says it will.)