When you think about it for the first time, you might be led to believe that burning the tree would be what kills the Night King.
However, looking carefully through the scene in the fifth episode of the sixth season, you will notice the door in the background, that one mountain shaped like an arrowhead. This shows the location to be north of the wall hence contradicting the other symbol which has been known to “The Gods Eye”, which was first shown in the very first episode of the first season.
The Gods Eye is said to have significant importance to the children of the forest and The Weirwood Trees. It is located northwest of the Kings Landing.
In speculation, there might be one main tree say, The Grand Weirwood. The Wight walkers may as well be obtaining their power from the Weirwood Trees since the children of the forest, who created the Night King are powered by the same.
To put all these together, the Wight that Beric killed represents the Night Kings power. The Weirwood in which the scene was centered shows the Grand Weirwood Tree which is on Gods Eye hence making it the primary source of power. To succeed in doing away with the Night King, one can guess that they will have to burn down the Grand Weirwood with a flaming sword. Jon might be the chosen to do so, with Beric being his teacher or instructor.
In the scene, the flaming sword with Beric will start traveling around the spiral outlets. This might be a sign that everything that draws power from the Grand Weirwood will eventually die with the Night King causing possible drawbacks.
There is a high chance that it was the children of the forest who also created the three-eyed raven as they created the white walkers.
It might have been a counter-measure once the children realized that what they had created previously was getting out of hand. The three-eyed raven was meant to be an all-knowing library that houses the memory of defeating the Night King. If this is so, it would mean that Bran would eventually die as well.
One can also speculate that the Lord of the light is somehow connected to the Grand Weirwood. Although they may be posing two different religions, they might have one source.
If the above argument is valid, then everything connected to magic, including Jon and Beric would die. This is not an ideal kind of heading so there are chances that Beric will die early. Also, Bran might have to die as well to be at a critical point of revealing to Jon how to use the flaming sword.
In the seventh season of Game of Thrones, there are two other clues which include fire as well.
In the first episode of the season, Hound sees something in the fire that later comes to pass as the wall coming down at the east watch of the sea. The other clue is on the sixth episode where a keen eye will notice the camera panned along with the carved Westeros table located on dragon stone. They were panning from the south of Westeros up to the north. In this shot, it conveniently had the fireplace in the background positioned directly north of where the Wall is carved on the table. This is later shown as the dragons start shooting fire north of the wall.
With these kinds of clues being dropped over and over, it may be correct to say that we have been given a clue as well on how the Night King will be killed.