To not waste your breath means to not bother trying to convince someone do do something.
You're wasting your breath by telling the person not to do something as they will ignore you and still do what they should not be doing.
For example if you tell your dog to not eat rocks but he will do so anyway.
Then you would be wasting your breath by telling your dog to not eat those rocks as the dog will eat the rocks anyway.
To waste one's breath means to say something that will probably be ignored or not have any effect.
So you should not use your breath to say that thing or argue with the person.
It's like the idiom or saying of beating a dead horse which means there will be no outcome in what you're arguing about or talking about to the person.
Some people are so stubborn and will do something they should not do even if you try to tell them otherwise.
That would be beating a dead horse or wasting your breath.