Should I use treated wood or not? Is it preferred to use treated lumber for the studs and plywood for a firewood shed?
Is there a reason not to use treated wood?
It probably depends on how much effort you're going to be putting into it.
If you're just planning on slapping something together quickly, then sure, go for untreated, and if something goes wrong, you can rebuild it in a couple of years.
I'd personally use pressure treated, as I'd have to make a larger structure due to local ordinances. (I can't store firewood within 18" of earth; I don't know if it's an attempt to deal with an invasive species that caused a ban on exporting firewood from our county) Because of the extra work involved, any savings on cheaper materials would be wasted if I have to rebuild.
I'd also be interested in the anti-insect properties of pressure treated, not only the anti-rot aspects.
Treated lumber is much more expensive than regular wood. Using regular lumber and painting it would be much more economical and will last just as long. Also treated lumber requires special fasteners that won't rust and that adds even more to the cost.