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.

This question would depend greatly on someone's individual experiences. To me the order from strongest to weakest is: 1, Save your breath. 2, shut up 3, Hold your tongue. However, 'hold you tongue' sounds like something from Lord of The Rings or Game of Thrones. If someone said this to me I would think they were being sarcastic. "save your breath" is used (in my opinion) when you are telling someone their excuse is not good enough (i don't know how to say this properly) or that they have said too much and it is offensive. "shut up" is pretty weak, I only put it at number 2 because it's more natural to hear someone say this as "please be quiet", at least how I was raised ;)

Shut up is really strong. Hold your tongue is not particularly used. It's grumpy old people's word. Save your breath is the softest of them. However, it's more condescending than vulgar, so watch out with it.