How does the soul-tie or the relationship between Roderick and his twin sister, Madeline Usher, drive the whole story in "The Fall of the House of Usher"?

The whole story leads up to the climatic ending in which brother and sister die together and the house almost immediately collapses into the tarn.
Roderick and his twin sister, Madeline, are the last remaining members of the Usher family, known as the house of Usher. Their line—the genealogical house of Usher—is somehow mystically and supernaturally tied to the physical house of Usher they live in.
Beyond that, Roderick and Madeline share a strange symbiotic tie. They are dopplegangers, the mirror images of each other, and clearly, as Roderick understands, he cannot live if Madeline dies. When she escapes from the crypt and falls dead at his feet, he too perishes in the same moment:

With a low moaning cry, [Madeline] fell heavily inward upon the person of her brother, and in her violent and now final death-agonies, bore him to the floor a corpse . . .

The strange physical decline in Madeline is mirrored in the nervous mental condition that plagues Roderick. Both characters are symbols of sickness, and the house, too, covered in mold and fissures, replicates that sickness.
This is classic Gothic literature: Gothic holds up a mirror of parts of ourselves we don't want to see. In the dying Madeline, Roderick sees his own dying, and it terrifies him. Likewise, Gothic literature shows the creepy side of family that we often don't want to acknowledge, symbolized in haunted or creepy houses like Usher that are a distorted version of the happy homes our conscious minds like to conjure.
Both Madeline and Roderick are joined by decline and a deathly sickness, and that drives the story to its gruesome and terrifying end.

The idea of the "soul tie" that Edgar Allan Poe puts forward definitely drives his story, but we must recall that the story has a dual filter. First, the story is told by the narrator, an old friend of Roderick who admits to not actually knowing him all that well. Second, Roderick is the sole source of the information that the narrator shares with the reader. We do not actually hear from Madeline so we cannot be sure that she shares the idea of being her brother's soul mate.
The theme of family lineage being tied to incest is key in two ways. Roderick is convinced of its centrality, and shuts his sister up while he contemplates the next step. But because he acted rashly, he actually killed his sister and thereby destroyed their line—their house, which crumbles around them.
So says the narrator.