It depends on the location, the farming operation, and the markets.
In wetter areas, stubble can be turned back into the soil as there is enough moisture for it to biodegrade rapidly. Burning can add potash back into the soil, IF you can get it turned under before the wind takes it away and gives it to somebody else.
In dry areas, things like corn and sunflower stalks are too big and coarse. It takes years for them to deteriorate into the soil. They have to either be burned, grazed off, or picked up.
If the stubble and other residue is baled, it can be used as part of a livestock feed ration, bedding, or mulch. If it is used locally, then the resulting manure can be put back on the fields in a form land in the more arid areas can use. Manure breaks down quickly and the nutrients are concentrated so it takes much less water and microbial action to turn it back into soil - it's already half way there!
The same is true for grazing stubble after harvest. Cows, goats, sheep and even horses can use the residue to grow and turn the extra into fertilizer the land can use instead of those big stalks that take sometimes a decade or more to break down.