From how I understand it heat will not simply disappear, it will only dissipate and equalize throughout all matter. If we had a closed system with a single hot object and a cloud of cold matter then after a period of time the heat would equalize between the two and they would remain at the temperature until something else changed the balance. You could think of it as a faucet (heat source) that is putting out some sort of thick gel (heat), as the gel falls out it piles up in a hill (high temperatures), but eventually spreads out to being level (equilibrium temperature). As long as you continue to put out gel there will be a hill, but if you turn it off and wait eventually it will be perfectly level.
As far as taking heat to cool, you can only even cool a local area at the expense of heating up another area at least as much as you cooled the first down (it would take 100% efficacy to be the same). Really you can only move heat from one place to another (and in the process add more due to inefficiencies), that's why AC and refrigerators have radiators outside the area that is to be cooled.