In S08E06 of Game of Thrones, we see Drogon burn down the
Why did he do that? Did he grow a conscience and see that it was a source of evil?
Lazy writing and symbolism (Daeron explained the symbolism in his answer).
A couple of in-universe explanations:
He was going to incinerate Jon but at the last second decided against it. He already generated the fire and had to release it, the Iron Throne was simply in the direction of his fire breath by coincidence.
Dragons are very smart, he realized that his mother died because she was chasing the Throne and burned it in rage.
After watching it again there were two separate fire breaths, Drogon first released the fire he was going to burn Jon with without harming the Irone Throne and then destroyed the throne on purpose with a separate fire breath.
This means the first option is wrong and leaves only the second.
Here's a nice little summary.
The red-tinted dragon then unleashes a powerful stream of flame aimed right at the very thing that has been such a point of contention since season one. He succeeds in melting all the swords down to a bubbling, molten puddle. Just as Tyrion says later on in the episode when discussing why Bran would make a good king — "Sons of kings can be cruel and stupid as you well know . . . that is the wheel our Queen wanted to break" — Drogon succeeds in symbolically erasing the idea of a monarchy, whether purposefully or not.
As to whether or not Monarchy is actually gone, in the quote above it is to be interpreted that an elective monarchy is not a monarchy. Here is Tyrion's quote.
(Tyrion) From now on, rulers will not be born. They will be chosen on this spot by the lords and ladies of Westeros to serve the realm.
Game Of Thrones, Season 8, Episode 8 (The Iron Throne)
As far as official statements go, there has been no explnation as to Drogon's intentions in melting it. It looked to me like he was trying to send Jon a message; if Dany couldn't have the throne, no one could.
Dareon's answer nails it in terms of symbolism. My guess on an in-universe answer is that Drogon has spent so much time linked to Dany, he knows the throne was the object that led her to her death. (Pure speculation, mind you!)