If he speaks, he’s afraid his voice will break.
Zuko is ranting, “You have no right to judge me,” as only a man judging himself would say. If Iroh speaks, his voice might break, and he doesn’t want this Zuko to hear that.
There had been another Zuko – seems almost like a dream now. A Zuko who smiled, and told Iroh he deserved to have the best tea shop in the city. Now that Zuko is gone, and replaced by one who screams that Iroh should be sleeping in the gutter.
Zuko storms out. Iroh doesn’t cry. Not this time.