The statement that one should not use a double negative is a caution against a particular dialect form well-known to native English speakers. It is something primary school teachers say to native speaking children. The caution should not be understood literally. It does not mean that all uses of double negation are incorrect.
A more precise statement of the rule would be: Standard English does not acknowledge negative concord. This dialect sentence displays negative concord:
If I don't use no microphone, no one will hear me.
Notice that in the first clause both the verb and its direct object are negated. This is negative concord because the two negatives are intended to reinforce one another.
As we have already said, standard English does not recognize negative concord. Instead double negatives cancel one another out to produce a positive. To illustrate this a teacher might ask the children to interpret the dialect sentence above as if it were standard English. The teacher guides them to an answer such as this:
I must not use a microphone if I want to be heard.
This is absurd of course. But it illustrates an important point: double negatives are not forbidden in standard English, they simple cancel one another out. It is only an error if negative concord was intended.
For example, this use of a double negative is good literary English:
Your complaints have not gone unheard.
The "not" and "un-" cancel one another out, so the sentence means:
Your complaints have been heard.
(Though the emphasis is a little different.)
The sentence in your question does not display negative concord like our dialect sentence. Nor do the negations cancel one another out (as in our example from literary English) since they are not in the same clause. Instead the two negations are in separate clauses joined in an if-then construct. We can simplify the sentence to this:
If no microphone, then no hearers.
We must use two negations in this sentence because the message is about two negations: negating the microphone negates the hearers.