Having a Duplicate allows you to use it to save him from any effect that would make him leave play (kill, discard, remove from game, return to hand... etc). However, there are specific cards - so far most notably Ghaston Grey, Marched to the Wall and Wildfire Assault, that do *not* allow you to attempt to "save" a character. They explicitly have the text informing you that the character cannot be saved from the effect.
So, while a duplicate can protect Tywin from military claim or an opponent's put to the sword, it does not protect him at all from Marched to the Wall.
Oh ok, got it. So it comes down to having either a 0-1 cost character in hand or the other guy not having the plot. Although down the line, plots that interact with each other and negate its effects might happen