Film crit pro-tip (say that five times fast): always be on the lookout for chairs. As Tony Zhou over at "Every Frame A Painting" says,

In film, a chair is not just a chair […] And the type of chair can say everything about the person and the world they inhabit. […] The first, and more common use of a chair is as the extension of the world.

And what world does King George VI occupy? The world of the dang English monarchy. Our Bertie is petrified of the power his position gives him, and the responsibility that he has as the public face of England.

This fear is never as apparent as in Bertie and Logue's final blowup argument, when Bertie tries to fire Logue because he lacks proper credentials (even those the properly-credentialed doctors he's been seeing have given him such sage medical advice as "Smoke a lot of cigarettes!") He barks at Logue, stammering all the while,

BERTIE: With war looming, you've saddled this nation with a voiceless king. You've destroyed the happiness of my family, all for the sake of ensnaring a star patient you couldn't possibly hope to assist. It'll be like mad King George III, only Mad King George The Stammerer, who let his people down so badly…

Logue is no fool, and he knows that Bertie's anger is an extension of his fear of being monarch. He also knows that he'll have to do something drastic to get Bertie's undivided attention. So he saunters over to Saint Edward's Chair and sits in it.

Now you might wonder what the big deal is—isn't a chair just a place to plant your butt?—but the fact is that only the newly-coronated kings and queens of England are supposed to sit on St. Edward's Chair. You can see Bertie's indignation in the way he turns and stammers,

BERTIE: That chair is the seat on which every King and Queen—

Logue is a sly dude. He know two things by now: a) Bertie speaks fluently when angry at Lionel and b) Bertie's true passions and allegiances come out when he's hot under the collar. So Logue cheekily says,

LOGUE: I don't care. I don't care how many Royal arseholes have sat in this chair.

But we know that the two men aren't arguing about a chair, though. Logue is trying to get Bertie so riled up that he yells fluently at the one person he trusts enough to be emotional around: his friend. And he's trying to make Bertie, through his anger, show that he actually cares about being king. He cares enough about being a good king that he's willing to break through his stammering and speak with passion.

So what does St. Edward's Chair symbolize (besides, you know, the crowning of English monarchs)? In the context of The King's Speech, it symbolizes the "tough" side of Logue's love for his buddy Bertie. He's willing to transgress pretty spectacularly in order to make Bertie realize that he cares about being king.