It symbolized the end of monarchy.
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.
The Meaning Behind Drogon's Fiery Symbolic Gesture in the Game of Thrones Finale - PopSugar
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)
Speculation: As far as official statements go, there has been no explanation 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.
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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.
Edit
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.
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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!)
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I watched this most emotional scene of the last season and in my opinion the reason is: he did not harm Jon, instead he burnt the Iron Throne because he knew that this throne is the reason behind his mother's death.
If he didn't know, then he would have burnt Jon.
The writer wants to tell the love of child for his mother.
No matter that Drogon is an animal; the love for the mother is same.
And the second case is: he knew his mother loved Jon, so he didn't harm him because Dany would not allowed him, if she was alive.
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Initially Drogon had the full intention to incinerate Jon as he felt the pain his mother did in his heart.
The unexpected bitter reality made him cry and he wanted to turn the reason and the person behind it to ashes.
Still he did not harm Jon, but he did burn the Iron Throne because he knew it was the throne Dany had been running after her whole life
and Drogon knew how much the throne meant to her
but when Drogon saw her dead, he burned the throne intentionally as he thought, if Dany could not sit in it, no one ever will.
The love of child for his mother is shown.
His love for his mother is very much like own.
After his brothers' death, she was the only one left to him.
And he guarded her for good.
He allowed Jon to go to her alone just because he knew she loved him, and he loved her and wouldn't harm her.
But as we know, Jon was the one to take her life.
The reason why Drogon didn't harm Jon is that he knew his mother loved Jon, and wouldn't see him harmed.
The other reason was that Jon had Targayren blood and dragons have been ridden by Targayrens for centuries.
So he did not dare kill one of his own.
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According to the script , Drogon did intentionally spare Jon, but the Iron Throne was just collateral damage as part of his rampage.
He looks down at Jon.
We see the fire build up in his throat.
Jon sees it as well.
He prepares to die.
But the blast is not for him.
Drogon wants to burn the world but he will not kill Jon.
He breathes fire on the back wall, blasting down what remains of the great red blocks of stone.
We look over Jon’s shoulder as the fire sweeps toward the throne– not the target of Drogon’s wrath, just a dumb bystander caught up in the conflagration.
We look through the blades of the throne as the flames engulf it, and blast the wall behind it.
However, just as Daeron's answer states, that doesn't mean that despite Drogon is NOT consciously aware of its symbolism, that it wasn't symbolically written this way for the story's benefit.
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In S08E06 of  Game of Thrones , we see Drogon burn down the    Iron Throne.   Why did he do that? Did he grow a conscience and see that it was a source of evil?
Why did Drogon do this?
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<plot-explanation><game-of-thrones>
movies.stackexchange.com/questions/100891
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The reason why Drogon didn't harm Jon is that he knew his mother loved Jon, and wouldn't see him harmed. According to the script, Drogon did intentionally spare Jon, but the Iron Throne was just collateral damage as part of his rampage. Drogon spared Jon intentionally, but the Iron Throne was an incidental casualty of his rampage. Writer wishes to discuss the love of a child for his/her mother. 
According to the script, Drogon spared Jon intentionally, but the Iron Throne was an incidental casualty of his rampage. Writer wishes to discuss the love of a child for his/her mother. In the second case, his mother loved him, so he couldn't have been harmed if she were still alive. Targaryens have ridden dragons for millennia, and blood has been in his family for generations, writer says. The writer says Drogan would have left Jon unharmed because he knows that his mother loves him, and doesn't want him harmed.