The loss is due to the difference in current that the wiring has to carry
All else being equal*, a 240V appliance will be more efficient than a 120V appliance, because:
Power loss in wiring = (current)^2 x (resistance)
To deliver the same amount of power to the appliance, 1/2 the current is needed in a 240V appliance, because:
Power to the appliance = (voltage) x (current)
Because the current is reduced by a factor of 1/2, the power loss in the wiring will be reduced by a factor of 1/4.
But how much difference does this make?
According to the U.S. Energy Information Agency's RECS study, the average U.S. household uses about 2000 kWh for air conditioning per year.
If we assume that the a/c runs 100 days out of the year for an average of 10 hours a day (1000 hours total), that means the average demand is 2000 W when it's running, so about 17 amps for a 120V a/c, or 8 amps for a 240V a/c.
Let's assume it's wired with 20' of 12 AWG wire in both cases, which has a resistance of 0.032 ohms (source).
Thus, total power dissipation in the wire is
17^2 x 0.032 = ~ 9 W for the 120V a/c. Over 1,000 hours that's 9 kWh, or $1.30 of electricity at the average U.S. residential rate. For the 240V a/c, dissipation is about 2 kWh, giving you a savings of $1.00 a year.
*Obviously, all else is not equal, the details of which some of the other answers get into. You'd be hard pressed to find two a/c units where the only difference was the operating voltage.