jim rozen writes:

What if you just take the tank and don't hook it up to
*anything*? That is, simply open the valve and vent it to
the atmosphere? Then you are pretty much at example (1) above,
but the second 'tank' is really, really large. So when the
surrounding environs heat up, the temperature rise is really
tiny - so small you don't notice it.

No. Compression and expansion do not create heat. If you vent a compressed
tank to the atmosphere, the energy is spent in three ways: (1) work
expanding the planetary atmosphere, which is not heat, and which is
theoretically recoverable, (2) friction heating at the orifice, and (3)
kinetic energy in the motion of the exiting flow, which kinetic energy
eventually turns to heat as the moving air settles.

If you were to take a tank of compressed air above the atmosphere and vent
it in the vacuum of space, then (1) is no longer present, (2) is about the
same, and (3) does not turn to heat, but the gas retains its kinetic
energy, moving outward and expanding forever.