You guys are both right -but looking at it in different ways.
If you pull 1 kWh from the electrical panel, it makes no difference if it's 110, 220 or single phase or 3 phase... The power company charges you for 1kWh. Plain and simple.
With 3 phase equipment and 220 single phase, the current draw on the conductors is reduced. When conductors carry current, there are small power losses in the wiring that is proportional to the square of the measured current. If you reduce current by 1/2 or 1/3 in each conductor, you decrease power losses in the conductors dramatically more. If you use properly sized wiring, the losses in the first place are negligible. What this means, is that with 220 single phase or 3 phase equipment, more of the power you're paying for goes into producing work and less is lost in heat in the circuit providing the power. Again, with properly sized wiring, the heat losses in the wiring is negligible.
Next: Because of the mechanics of a 3 phase motor, they are inherently more efficient by a few percent. 3 little pushes on the rotor is more efficient than one big one. A few percent is not a lot. BTW: Mechanically, there is no advantage in single phase 110 vs 220 -both spin the rotor with one big push every rotation. 3 Phase pushes 3 times with every complete rotation and it is a mechanically superior design.
Also, single phase motors (110 or 220) need start and possibly run capacitors. There are small losses in the current flowing through a run capacitor -but not much. The start capacitor causes the motor to fight itself at startup and this causes a brief momentary period of high inefficiency at startup only. 3 phase motors do not need start capacitors or start windings.
As soon as you power a 3 phase motor with a VFD, all the inherent efficiencies get eaten-up by the VFD conversion process and circuitry. It's a wash.
Forget a minute about motors and think about a heating coil.
If the heating coil is putting out say 10kW and if the conductor wires supplying power are properly sized, it makes no difference if it's 110, 220 or single or 3 phase. It will cost exactly the same out of your wallet. If the wires are not properly sized, it will cost more to run the single phase than for the 3 phase.
Ray