I fed my ducks sweet potato vines last year with no problems. And I would throw in a couple pounds of the vines at a time after harvesting the tubers. Sweet potatoes leaves, while in the morning glory family, are actually edible for humans as well. I had them in a few salads and used them as a replacement for lettuce in the heat of summer when lettuce typically doesn't do so well. Just as a personal preference I use the younger leaves as they tend to be less papery and tough.
Before I fed my ducks the vines I scoured the internet and found some study where sweet potato vines were used in a field experiment in Africa as an alternative to traditional feeds. From what I can recall, the sweet potato did not work as well as traditional, but since I'm not exclusively using them I figured it couldn't hurt to supplement with them.
Also, normal potato are in the nightshade family and the leaves/vines are toxic in sufficient quantity.
I cook the sweet potatoes for myself so the chickens only get the skins and left-overs. I cook the whole potato so they are eating cooked skins. I also cook squash for them from time to time. They love those, too.