I suspect something Bran does will bring down the wall.
Bran is “marked” by the Night’s King. It was that mark that broke the spell of protection in the Three Eyed Raven’s cave. I think that same mark will break the spells that protect the Wall when Bran passes through underneath.
They pretty much set it up that he’s going to at least allow the White Walkers to follow him through the Wall if he passes through it with that mark on him.
In the books there is also the “horn of Joramun” which supposedly can bring down the wall, but they didn’t really introduce that on the show.