Chas,
I'm not a metallurgist, or a materials scientist, but will go so far as to say, pretty much everything solid is a spring, and you could stretch the point to say that if sound travels through it, it’s a spring of some sort, so gases and liquids too. Some things are far worse than others, because they are really soft (eg solder wire, or jelly) or reach a failure point before they can do anything useful (brittle things like glass) but pretty much everything deforms when you push it, and springs back to where it was when you stop. Even glass, it “Pings” when you tap it, the vibration is a result of its springiness. Basically, if it deforms a bit under load, and returns to its previous shape when the force is removed…
“Springs” as in bits of coiled wire, or leaf springs, can be made from various varieties of steels, alloyed normally, stainless sometimes, or bronzes, and other materials, including plastics & rubbers. it’s generally true to say that useful springs are made of hard materials that are not too brittle, nor too ductile. Cheese is particularly useless.
If we’re talking steel, you can soften and re-harden it with temperature, quenching, slow cooling, etc, but it’s likely that, unless you have very precise control over your processes, you’re going to degrade it so you’d probably choose not to do it more than necessary. Lots of stories about making things that need to be ultra hard (cutting edges) and yet not brittle, samurai swords being a famous example.
Bronzes are generally not treated in the same way as steels, and I don‘t believe that tempering is possible, but please refer to my opening statement!
obviously none of this applies to plastic or composite springs (think fibreglass bow for archery, or some vehicles which have composite suspension springs) where heat treatment is not part if the process of manufacture.
I’m not at all sure this is a useful answer to your question…