Why are wood burners painted black?

My guess would be to render any smoke/soot that gets out and on the stove invisible.... you'd get that nasty black smudge if you touched it but visually you wouldn't be able to distinguish the soot from the stove.... so I think its purely cosmetic

answer 2 Well, the answer is in the question "... painted ...".

If you choose an enamelled stove finish, then colour is no problem.

The ordinary pigments in paint are not resistant to heat. [Though my woodburner has decorative panels of red.]

Older coal-fired kitchen ranges were coated with refractory enamel, (also called vitreous enamel) which lasted for decades. So it need not be re-invented - just pay the money.

another idea

Most stoves are made of cast iron, which is naturally black itself, unless it rusts.

You might be referring to stove black. This is a product available in hardware stores, and it is made by many different manufacturers. It is a protective finish that can be safely applied to cast iron wood stoves.

You really cannot safely paint a wood stove, as almost any paint will decompose, smoke, and possibly catch fire.