The original 12 volt powered circuit uses a 15 ohm current limiting resistor in series with the IR LED and because the LED has a forward volt drop of 1.28 volts at 100 mA, I believe that the 15 ohm resistor could be too low for 12 volts.
Basically it works like this - When the transistor is fully turned on it might drop 0.2 volts across collector and emitter leaving 11.8 volts across IR LED and 15 ohm resistor. Because the IR LED can be assumed to drop 1.28 volts, this leavesabout 10.5 volts across a 15 ohm resistor i.e. a current is implied of 700mA.
This is far too much for the LED and it will burn so we then have to assume that the voltage applied to the base via the 2.2kohm resistor is maybe about 1 volt max. This means that maybe 0.3 volts is dropped across the 2k2 base resistor implying a current into the base of 0.136 mA. If the transistor has a gain of (say) 100, the current flow in the collector is going to be about 14mA.
But this creates an issue in trying to get the circuit to work from a 3V button cell. With maybe 14mA flowing through the transistor, 15 ohm resistor and IR LED, the transistor is dropping about 10 volts across its terminals and this won't be as clear-cut on a smaller supply voltage.
The upshot of all my ramblings is that without knowing the precise details of the input voltage feeding the 2k2 resistor it will be guesswork to predict if the circuit will work from a 3V supply.
EDIT - as T Crosley mentions, a button cell will only be capable of supplying a handful of milli-amps (even the bigger ones). See the data sheets on this page for proof.