You can use treated wood for the part that touches the ground, if you really want to. However, completely aside from any possible toxicity and environmentalism issues, treated wood is HEAVIER than regular wood, so since weight is always at a premium in a tractor, you want to use as little treated wood as you possibly can.
If pressure treated wood is in good shape there is no reason to think it will harm chickens (chickens do not peck/chew sound wood), especially in a tractor where it's not like it's constantly leaching into the same bit of soil where bugs and grass grow from. But, it is a personal choice. Cedar and redwood (and locust or osage orange, if you can get them, which you probably can't where you are) are not as long-lived as pressure-treated, but with good selection and painting/sealing they can last a fair long time, even in ground contact.
For the majority of your tractor, just regular ol' spruce-pine-fir is perfectly fine, assuming you prime and paint it well (multiple thin coats, not fewer thick coats!). It is a delicate balance between sturdiness and weight.
Good luck, have fun,
Pat