Even if the energy to run an air conditioner came from a source like hydroelectric power and the machine operated at 100% efficiency, it would still heat the outside air.

A typical air conditioner is one example of a heat pump, a device that moves heat from one location to another. In this case you're pumping heat from inside your house to outside, warming the air around the house.

Your refrigerator is also a heat pump, moving heat from inside the fridge to the outside. This is one reason cats like to climb up on top of the refrigerator (besides the fact that they like to climb on everything) - it's warm up there.

https://en.wikipedia.org/wiki/Heat_pump

It's less about heating the outside air directly and more about carbon emissions from massive cities all using air conditioners at once.

Not saying give up air conditioning altogether, but consider hanging out in an air conditioned public space instead of switching on your own. Libraries, malls, coffee shops, universities, just to name a few.

From the basic energy conservation considerations, it is clear how much heat is produced by the AC - the same amount that it consumes from the power grid. There is no other place that energy can go to, and there's no additional energy that can be converted to heat. So if your AC consumed 1kWh, that's 1kWh of heat that was dissipated. It doesn't matter what's inside the AC, conservation of energy works the same way. BTW the same would be true for any device that consumes 1kWh, unless it creates mass (very uncommon for household devices).

It would heat the outside air only temporarily. Eventually the heat would seep back into your house and it would be back like it started.

Of course 100% efficiency isn't actually possible.