Yes and no. Correct English does not use a double negative, but when it is used it can either reinforce or convey the opposite... which is why we don't use double negatives, it's confusing and language should be clear and precise when used properly.
What if someone says "no no" or "no dont". That conveys a negative response despite being two negatives. Granted these aren't proper English or full sentences, but it's something you might say
"No, don't [do that]" is not a double negative, it's just two independent negatives in a row, they don't affect each other. A double negative would be like "don't not do that" which will mean "do not fail to do that" to most people.