will the chemicals in the pressure-treated wood have any effect on the hops?
Good question. I don't think so.....
No, the latticework won't but pressure treated post will for a few years as the chemicals leach out. Just don't plant anything right next to the post, say 8", so the roots aren't slurping up the chemicals and you will be fine. "Weathered" or raw post are better. The latticework in contact with the bines won't have any effect.