Yeah, I definitely wouldn't feed any part of a potato or sweet potato to rabbits (including the potato chips that Layne swears we need to share with him).
I hadn't actually heard the unripe/ripe potato distinction before... for sugar gliders (we've got a cute pair of 'em), the safe/unsafe food lists specify raw vs cooked (with raw being toxic regardless of if they're ripe or not). Either way, raw potato is a safety gamble at BEST. Even cooked, both sweet and regular potatoes are starches - chock full of carbs. Carbs in general (NOT just sugars) elevate blood sugar, raising the risk of diabetes when eaten in excess.
"In excess" doesn't take a whole lot for a rabbit because their natural diet is very low carb and they're more prone to diabetes than other pets. Rabbits do have a major sweet tooth and a wild rabbit would gorge itself on fruit if it was in season and accessible to ground level... but a wild rabbit would most likely succumb to illness, injury or predation before it was old enough to develop diabetes. In captivity, our pet rabbits live significantly longer, making it especially important to keep them on a high (indigestible, like from hay) fiber and low carb diet.