When we say "Don't waste your breath", we mean "Don't bother saying anything, it's not worth the breath that you will use expelling the words from your mouth".
If I keep saying to a child "Don't stick your finger in that electric plug socket" and the child keeps sticking its finger in the socket, I would probably give up and say "I'm wasting my breath. S*d it. Let him electrocute himself!"
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