The use of 'no' in both of those sentences is incorrect. 'No' being paired 'don't' makes a double negative. It should be "We don't need education" and "I don't have money."

The 'no' is placed in these sentences incorrectly. For both of these sentences the second option is correct. Because there is already a don't in the sentence there is not a need for the 'no'. If you have no and don't in the same sentence, usually this is a double negative. In English you should not use a double negative ideally but you often hear some people using the.

You could also say: We don't need any education.

In the first, it's supposed to be somewhat humorous. In school, that construction would be corrected to "We don't need an education" (or something else that got rid of the double negative, which is not a part of Standard English) so saying "We don't need no education" is supposed to show that the speaker does actually need an education. It's a weak joke. Generally speaking, though, those phrases ("don't got no" or "don't need no") are associated with non-Standard English dialects. In what you're probably learning, the "no" doesn't belong in the sentence.

These examples are usually slang. "I don't need no money" "I don't need no education" - slang (incorrect) I don't need any money. I don't need an education. - correct.