When assessing the conflict(s) in a story, it can be helpful to identify its climax, or the moment of the most tension in the text. It can also be a turning point of some kind. In this story, the climax takes place when Madeline Usher emerges from her tomb, still alive (though her brother and the narrator believed she was dead when they interred her) and covered in blood, looking wild and frightful. She approaches her brother and falls, dead, on top of his body. He falls to the floor underneath her and dies in this moment as well.
It's a strange moment that contains a weird intimacy, but these two people have always been close intimates, so to speak: they shared a womb (as twins), and they were also expected to continue the Usher line by having children together before Madeline grew sick. The narrator points out that there are no offshoots on the Usher family tree, and it is, perhaps, this pattern of mating with one's siblings that has caused the illnesses that have corrupted the physical and mental health of Roderick and his sister.
For this reason, I think it makes the most sense to identify the main conflict of the story as a character vs. character conflict (between Roderick and Madeline), but we might also consider the possibility of a character vs. society conflict (between Roderick and his family as a result of their strange and skewed values: keeping the bloodline intact, etc).
The central conflict that is present in Edgar Allan Poe's "The House of Usher" involves the unstable conditions of Roderick Usher.
In this Gothic thriller, Roderick Usher suffers from internal conflicts. These conflicts are composed of three components: his sensory sensitivity, his fears, and his anxiety. Although the terminology was unavailable to Poe in his time, Poe describes in Usher the psychological condition of hyperesthesia, which is an extreme sensory sensitivity. For instance, he is extremely affected by such things as light, sounds, smells, and tastes. This "constitutional and family evil" involves Roderick's suffering from "a morbid acuteness of the senses":
...the most insipid food was alone endurable; he could only wear garments of a certain texture; the colors of all flowers were unendurable; his eyes were tortured by even a faint light.
There are also sounds that seem "peculiar" to Roderick. Some of these sounds are made by stringed instruments, and their vibrating sounds cause Roderick to become fearful.
As his fear grows, Roderick becomes terrorized by his acute anxiety. Because of this anxiety, Roderick has frightening anticipations of going mad and dying. In fact, as the narrator makes efforts to relieve his friend's fears and melancholy by engaging in painting with him and by listening to Roderick's dirges and unusual interpretive renditions of the music of Von Weber, he perceives:
...a full consciousness on the part of Usher of the tottering of his lofty reason upon her throne.
That is, Roderick Usher himself realizes that he is becoming mentally unstable. Usher's instability also becomes apparent as he tells the narrator that the old house itself is making odd sounds.
Roderick also suffers from hypochondria, the fear of illness. The Usher family has virtually wasted away, and only he and his sister remain. Roderick is especially worried about his sister's illness and the fact that if Madeline Usher dies, his only remaining relative will be gone. He is also probably anxious about his own health since she is his twin sister.
As he tries to cheer Usher by painting and reading with him, the narrator specifically alludes to hypochondria in his comments upon Roderick's painting:
...there arose out of the pure abstractions which the hypochondriac contrived to throw upon his canvas, an intensity of intolerable awe.
These unstable conditions of Roderick Usher contribute to the internal conflicts which develop the strange narrative and its horror in Poe's Gothic tale.
There are several conflicts in "The Fall of the House of Usher." First, there is the internal conflict that Roderick Usher is experiencing. He is losing his mind and his health, and he knows that his paranoia is growing. As the story progresses, Roderick is also struggling with the knowledge that he has essentially buried his sister alive, and he is waiting for her to emerge. This conflict between Roderick and Madeline is the main external struggle. Madeline's return is the end of both Roderick and Madeline, thus bringing the family Usher to an end.
These two conflicts are mirrored in the natural conflict seen in the short story. The weather is extremely significant in the story. As Roderick progressively loses the battle with himself and his sister, the weather progressively grows more and more severe, battering the already crumbling house. Just as Roderick loses his battle, and both he and Madeline die in the end, the house loses its conflict with the weather and literally falls. Poe effectively concludes with both the literal and figurative "houses" of Usher falling.