I have been told that green (unripe) potatoes are poisonous and that this is "common knowledge". I doubt there are any chemical properties in these potatoes that could cause any harm. AFAIK, potatoes are only a tuber that grows over time, gaining size to store greater amounts of (the same types of) nutrients.
With fruits, the maturation process makes sense because it's a "strategy" to spread the seeds (when they are ready and not before). Tubers are hidden in the ground, they are meant to store nutrients (as seeds) and not to be eaten to spread any seed (as fruits). However, as a part of the "intention" of not being eaten they may contain chemicals and their density may decrease as the tuber grows, so this could be possible.