Depends how much you like painting and re-building the trellis...
Pressure treat will last alot longer and won't need painting for at least a year. Then just a sealer will keep it up for a while. Regular wood HAS to be painted if you expect it to last for any reasonable amount of time.
Unless it is something like redwood, cedar or teak. I built trellises decades ago out of A grade redwood and out of teak that weathered gray but still have no real need of paint or stain save for esthetics. I raced my favorite vintage sailboat in salt water and its teak deck never had any sealer on it. It's outside all year round.
I would use treated lumber for anything in the ground though.
Personally? I don't like the natural gray weathered look and tried to get clients into a nice color of transparent stain at least.
If you start staining or painting outdoor decks or structures, it should be added to your annual or bi-annual maintenance list. The wood will hold its own but the stain or paint will not.