Why did Drogon burn the Iron Throne?
Let's get one thing out of the way: Drogon burned down the Iron Throne because it looked cool. The first 40 minutes of the Game of Thrones series finale possessed a number of images with a horrific, over-the-top potency -- Tyrion discovering his brother's golden hand in the rubble, dragon wings appearing behind Dany's back as she appeared on the stairs, Drogon emerging from under an enormous pile of ashes as Jon Snow approached the throne room -- but the sight of Drogon laying waste to the Iron Throne, the show's terrifying symbol of monarchy and warfare, was as striking as the series has ever been. In their first episode as directors, show-runners David Benioff and D.B. Weiss weren't afraid to unleash hell.
Besides the visual spectacle, the moment also had thematic power to it. On some level, Drogon clearly knew that despite the fact that Jon Snow just killed his mother, the former King of the North was not the real enemy in this situation. With its strong grip on the imaginations of the leaders in Westeros, the Iron Throne itself was the source of so much of the suffering, murdering, and scheming that drove the series. Instead of arguing that individuals were inherently evil or cruel, the show presented a world where good men and women acted against their better instincts in the hope of wielding the chair's power. Just look at the awed way Daenerys approached the chair, like it might start levitating or shooting sparks at any moment.
But, in the end, the Iron Throne didn't have any innate power or mystical qualities. It was just a fancy piece of furniture, a fact that Drogon affirmed by melting it to the ground. (The moment was slyly echoed later in the episode when Tyrion, back in his old role as the Hand of the King, spent time fiddling with the chairs before his first big meeting.) While it's possible to speculate that Drogon was consumed with grief in that moment and just wanted to set something ablaze, I like to think he knew he was helping to destroy the system that turned him into a winged instrument of death.