My dog is 1/4 lab and has already uprooted 3 newly sprouted rhizomes this spring. Thankfully she didn't think they were sticks or she would have eaten them. I replanted and I think they're ok. There is chicken wire going around my trellis now, so that's the end of that.
I doubt she would try to eat a cone if it fell on the ground, as she won't generally touch vegetable matter as food, but if they smell strongly, I can see her rubbing her neck on them like she does with any other stinky thing... rotten fish, bird droppings, things like that. I'd rather she smell like hops.