Lets say I got an item thatll run on 120v or 220v.

It pulls 10 amps @ 120 and 5 amps @ 220v.

I can wire the outlet either way, without issue.

But the question is, which one will keep the power bill down?

PS - Me and buddy got to pondering about this, and we couldnt come up with a valid reason either way, hoping someone has more insight. Everything hypothetical

The amount of power the device uses is independent of the voltage it runs at.  Or at least close enough that you'll never see the difference in your electric bill.

Thats kinda what i figured. The only advantage to running 220v would be I could use smaller wire to do so?

neither will keep the electric bill down as the power (1100W) is constant

220v will incur less % voltage drop and should run more efficiently.

in the real world, that means your item will come up to speed faster (ie: motor, saw, etc), and the lights won't dim when you turn it on...

3 phase power is where electrical savings can come into play.  This is mostly on real high demand items such as large motors.  Google "Power Factor for Three Phase Systems".

Depends on the outlet do not wire a standard duplex rec rated for 120v to 240

I worded that wrong, I meant if it wasnt a hyptotheical question, I would be adding a new circuit

The 240 V will have lower wiring losses (I^2R, from the current (I) flowing in the sire) and you are paying for those also.

While the motor will deliver the same power, reduced current decreases the other losses you are also paying for (the meter measures real total power, Watt-hours)).

3-phase systems have no special advantage in power factor, except that the POCO charges more for poor power factor and less for better power factor since ht distribution equipment must be based on the V-A load, and a poor power factor means larger distribution equipment without an increase in billable dissipated power (W-hr).
3-phase power often operates at significantly higher voltage, and hits reduces current (and current induced losses) for both the used and the POCO.

220v will have a more stable current that WILL save you money.

Given the option,i'd run it at 240v.
This will keep your service balanced.

Brickeye is correct, but my contribution is that there would be no change in wire size as a branch circuit would require a 15A breaker and 14 gauge wire.