As a problem too big for small minds, fuccboi obliviousness seems to regenerate like an unstoppable walker whenever you try to cut it down. Just like A Song of Ice and Fire, it may be that only dragonfire can melt the icy armor of a Night King-level fuccboi.
We know that a lot of dragons died during the Doom of Valyria, the cataclysm of an unspecified nature which did contain a lot of fire.In The World of Ice & Fire, there is a picture of dragon with fire on his back with the description A dragon burning during the Doom..It doesn't say this dragon died because of the fire, but it does prove that dragons can burn (unless we attribute this to the unreliable narrator).
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So was the fire during the Doom of Valyria simply much hotter than regular fire, and is that the reason it could hurt dragons?Or did the fire during the Doom of Valyria not hurt the dragons, but did they die of other causes (e.g. debris flying around)?
See, the dragons are immune to fire, up to a certain point. In general, the younger they are, the less resilient they are, also their own fire is weaker in comparison to older dragons.
The Dance
It is probably safe to assume that the dragons involved in The Dance of The Dragons were not affected by each others' fire. Most of them were fully grown and extremely old dragons - the likes of Caraxes, Meraxes, Vhagar, Sunfyre and Balerion the Black Dread.
History
It is true that generally [Targaryen] dragons have died a violent death, this is especially true given that they were used extensively for war purposes, but there has been no known occurrence of dragons burning each other to death during any of these conflicts.
Although, there are some clear examples of dragons burning each other, as exemplified in the answer privided by PrisonMonkeys; killing one is a whole different story.
The Phrase
Logically, many people, especially Targaryens themselves attribute themselves to having dragon-like properties, which includes immunity to fire. This clearly stems from the fact that if dragons can't be burned, and Targaryens think that they are dragons, then they also must be immune.
This is where Dany's mistake comes in; she is repeatedly taught that Targaryens are dragons, etc, by her idiotic1 brother, Viserys. This makes her think that she is immune to fire2; when in-fact they are not.
This is a description of the battle at Dragonstone between King Aegon II on Sunfyre and Princess Baela on Moondancer.Although to me it's a bit unclear which dragon blinded the other, it's a clear example of one dragon harming another with its fire.
You are asking about a fire during the Doom of Valyria hotter than regular fire, and about something burning a dragon on the picture called "A dragon burning during the Doom" so I remind you it can possibly be Wild fire, the GOT version of Greek Fire.
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