Only the green bits of potato contain the toxin, the production is induced by light. White/yellow potatoes mainly contain starch, that should not be dangerous. I've never fed my bunnies potatoes, though.
Found this on Wikipedia:
Solanine in potatoes
Solanine occurs naturally in all nightshades, including tomatoes, Capsicum, tobacco and Eggplant, as well as plants from other species. However the most ingested solanine is from the consumption of potatoes.
Potatoes naturally produce solanine and chaconine, a related glycoalkaloid, as a defense mechanism against insects, disease, and predators. Potato leaves and stems are naturally high in glycoalkaloids.
When potato tubers are exposed to light, they turn green and increase glycoalkaloid production. This is a natural defense to help prevent the uncovered tuber from being eaten. The green colour is from chlorophyll, and is itself harmless. However, it is an indication that increased level of solanine and chaconine may be present.
Some diseases, such as potato blight, can dramatically increase the levels of glycoalkaloids present in potatoes. This is believed to be a natural reaction of the plant in response to disease and damage.
Commercial varieties of potatoes are screened for solanine levels, and most have a solanine content of less than 0.2mg/g. However potatoes that have been exposed to light and started to green can show concentrations of 1mg/g or more. In these situations a single unpeeled potato can result in a dangerous dose.
[edit] Avoidance
Solanine and chaconine are present in potato shoots. In potato tubers 30–80% of the solanine develops in and under the skin and thus may be removed by peeling and removing the eyes. This is advisable if the tubers show green, but is not a guarantee of safety. Potato greening strongly suggests solanine build-up although each process can occur without the other. A bitter taste in a potato may be a more reliable indicator of toxicity.
Deep-frying potatoes at 170° C (306° F) will effectively lower glycoalkaloid levels, but microwaving is only somewhat effective and boiling is not.