So, why didn't dragon fire kill the Night King? Well as we heard the Three Eyed Raven say in the previous episode no one had ever tried it so there is no evidence it would ever work. Add to that, that the Night King is something more powerful than the White Walkers themselves and things go differently. For all we know the White Walkers could be killed by it and it is just the Night King who is immune to it.
As @Virusbomb has said we've also seen fire part when the White Walkers come near so whilst dragon fire is stronger than normal fire we could draw a parallel here too.
It's worth noting that the weapons we know of that can kill White Walkers are dragonglass and Valyrian steel and they are not forged in dragon fire. Thereofre, we can't draw any relationships between these items to make a theory.
Dragonglass is forged in the fires of the earth i.e. it is volcanic in nature.
"Obsidian," Maester Luwin insisted, holding out his wounded arm. "Forged in the fires of the gods, far below the earth. The children of the forest hunted with that, thousands of years ago. The children worked no metal. In place of mail, they wore long shirts of woven leaves and bound their legs in bark, so they seemed to melt into the wood. In place of swords, they carried blades of obsidian."
A Game of Thrones, Bran VII
Note that the art of forging Valyrian steel was lost in the Doom of Valyria, though some smiths can still rework it.
"Nor I, my lord," said the armorer. "I confess, these colors were not what I intended, and I do not know that I could duplicate them. Your lord father had asked for the crimson of your House, and it was that color I set out to infuse into the metal. But Valyrian steel is stubborn. These old swords remember, it is said, and they do not change easily. I worked half a hundred spells and brightened the red time and time again, but always the color would darken, as if the blade was drinking the sun from it. And some folds would not take the red at all, as you can see. If my lords of Lannister are displeased, I will of course try again, as many times as you should require, but—"
A Storm of Swords, Tyrion IV
However, it is mainly said to be forged with spells rather than dragon fire.
The blade was Valyrian steel, spell-forged and dark as smoke. Nothing held an edge like Valyrian steel.
A Game of Thrones, Bran I