This is based on the mention by Benjen Stark that the Wall is protected by magic, an additional measure to prevent the White Walkers and their undead wight minions from entering the Wall. So, it works similarly to the magical wards which sufficiently protected the Cave of the Three-eyed Raven from the White Walkers until Bran was "marked" by the Night King via greensight. Because of this, many viewers believe that the mark on Bran's wrist, which successfully cancelled out the protective spells in the cave, will spell the Wall's doom and facilitate the entry of the White Walkers into the realms of men.
However, there are a number of holes in this theory:
A corpse of a Night's Watch brother was brought into Castle Black in the Season 1 episode The Pointy End. The wight, named Othor while he was alive, was re-animated by the White Walker necromancy, even though it was in Castle Black, and effectively behind the barrier of magic that allegedly exists within the structure of the Wall. Jon burned the wight and the corpse of Jafer Flowers as well. In fact, it is a long-established custom in the Watch to burn their dead. Perhaps due to the knowledge of the White Walker's necromancy. So either the weakness of the spells in the Wall is due to lack of maintenance, or it's effects are simply too limited for such a structure covering more land area than the Cave of the Children of the Forest, which is all the more reason for the Wall, which is 300 miles long and 700 feet high, to be as large as it is.
The wights chased Bran and Meera through the woods after they raided the Cave of the Three-eyed Raven. If the mark on Bran's arm was a measure by the Night King to sabotage the Wall, why would the wights be required to hunt them down?
Benjen Stark was revived by a shard of dragonglass inserted into his heart ny the Children of the Forest. This is similar to how the Night King was created 8000 years ago. Season 6's The Door also demonstrated that the Night King was greenseer. Benjen implied in the following episode that he was aware of the contents of Bran's visions, which would mean he had some form of greensight. So in rescuing Bran, Benjen of all people should know what the mark would do to the Wall IF this theory was true. And yet his very actions would appear to contradict the idea.
Hodor's sacrifice in The Door would have been ultimately pointless if the Three-eyed Raven knew Bran's flight to the south would cancel out the magical wards inside the Wall.
Bran inherited all the knowledge of the Three-eyed Raven, so perhaps he can find/already know a method to remedy the mark and its effects, should it prove to be a continuing problem in the future.
That’s the multi-million dollar the GoT story question, what’s going to fault the Wall. A structure of 300 miles long and 700 feet high that built stoned, that covered depth ice, and ancient spells carved into the foundation of the Wall to prevent the dead from passing through...
Question, these “dead” are relate wights or does they include the Night King and the White Walkers – are they dead or have created with dragonglass that have half-life, like Benjen?
Wights are the undead. The zombie Night's Watch and Wildlings, and the re-animated skeletons. White Walkers are not undead as such. George R.R. Martin describes them as being "a different kind of life".
Only the Night King and Benjen Stark are the way they are because of enchanted dragonglass. The rest of the White Walkers are so created by the Night King's own touch. While the undead are animated by the sheer will of the Night King. As we saw in Season 5's Hardhome episode, the Night King merely made a gesture to re-animate the bodies of the freshly-slain Wildlings in the camp.