Two different answers in one plant
Sweet potato asks to be split into two questions, because the plant gives a rabbit two very different foods. The orange flesh is not poisonous — a rabbit who steals a small piece will come to no immediate harm — but it is one of the starchiest, sweetest vegetables in the kitchen, bred over centuries to be an energy store. That profile is nearly the opposite of what a rabbit's hindgut is built to ferment, which is long, low-energy fibre. The leaves and trailing vines of the same plant are another matter entirely: they are a genuine leafy green, comparable to other safe forage leaves, and a rabbit can eat them as part of the normal greens rotation. So the practical answer is: feed the leaves freely in rotation, treat the flesh as a rare curiosity, and never feel the vegetable itself is a necessary part of the menu.
If you offer the flesh at all
Keep it raw, keep it tiny, and keep it rare. A cube the size of a thumbnail, once in a while, is the scale — offered after hay, not before, and in place of that day's fruit or root-vegetable treat rather than alongside them. Never offer cooked sweet potato: roasting or boiling breaks the starches down into a softer, sweeter, faster-fermenting food, and cooked leftovers also arrive carrying oil, salt or spices. Rabbits who are overweight, or who have a history of soft droppings and uneaten caecotrophs when sugars creep up, are better off skipping the flesh altogether — they lose nothing by it. And if a rabbit raids a storage basket and eats a large amount of raw sweet potato, expect a day or two of digestive grumbling: push hay and water, cut all treats, and let the gut rebalance.
The leaves deserve the spotlight
Sweet potato leaves deserve better than being the forgotten part of this question. Gardeners who grow sweet potatoes often have vines in abundance through the warm months, and those leaves — picked fresh, washed, and served as part of a mixed handful of greens — give a rabbit variety that costs nothing and fits the diet properly. Introduce them gradually like any new green, watch droppings over the first couple of days, and then let them take a normal turn in the rotation. A word of caution on identification: feed leaves only from plants you are certain are sweet potato, grown without sprays; ornamental sweet potato vines from garden centres may have been treated, and wild look-alike vines should never be gathered on a guess.
Building the menu around leaves, not starch
The sweet potato lesson generalises: a rabbit's menu works when leafy, fibrous foods are the plan and starchy foods are footnotes. Writing that plan down is what keeps it true in a busy house, and in KinsLe's Care Teams the rotation can list the greens — sweet potato leaves among them on their days — with the treat items marked as the rare events they are, each one ticked off as it is served.