According to Nassim Taleb, when all the lions were killed in the Middle East, the goats multiplied, and they apparently love tree roots, which led to the Middle East turning into desert (not sure if it's true, but he says it is and I'm inclined to believe it).

Does anyone know what could be some unintended negative consequences of exterminating all snakes?

It always amazes me that in North America, they exterminated Wolves from most of place, and so were able to create a safe working environment for farming. Although the deer population is a little out of control, it's not a as bad as goats turning the land into desert.