It's an example of a double negative:
We do not need no education. <two negatives>
Prescriptive grammarians - grammarians that follow the traditional rules of the grammar - (would) argue that it's semantically awkward: "no education" means, education doesn't exist, so how can it be that "we don't need" something that doesn't exist?
Descriptive grammarians - grammarians that describe how speakers use language - (would) argue that certain double negatives serve to add emphasis: "no" adds further emphasis to "do not".