I think they all have different meanings and usages.
I would use "hold your tongue" when you want to say something, but you know it is better not to.
For example, your boss is going on and on about how wonderful your colleague's idea is, and you know that your colleague got that idea from someone else. But you know that now is not the time to tell your boss that.
"save your breath" is used as Matty Sue says (above answer) but also in other contexts. "Save your breath" = "you're wasting your time."
So for example, Sam thinks that Candidate A is wonderful. You think Candidate A is an idiot. You try to convince him, and your friend Karen says to you, "Save your breath. You'll never change Sam's mind."
"Shut up" is just "be quiet" in a rude way.