Tolkien was a devout Catholic. He claimed that Middle Earth was compatible with Christianity but original sin is a huge part of Judaeo/Christian beliefs so where does that fit in? We never find out about men. They awoke in the East and refused to talk about where they came from. There were dark implications but nothing definite.
The elves are a different matter. We know what their sin was. Or sins because they did it twice.
The world was shaped by the Valar and the Maya but Iluvatar created the elves and men (and Valar and Maya) and he decided where and when they would awake.
When the elves first awoke they were innocent. The Valar were overjoyed to find them and invited them to live with the Valar in the Undying Lands in the West in their city of Tuna. Some elves decided to remain in Middle Earth. Some started to follow the Valar but lingered and were left behind. Some made it across the ocean but decided to live on the coast and become ship-builders and sailors. But a substantial number followed the Valar to Tuna.
Then Feanor was born. He was the greatest of the smiths. He created the palantir and other miraculous gems. His greatest creation were three stones called the Silmarils which captured the light from the two trees. That's where the trouble started. Feanor began worrying that someone would steal the stones. At the same time the Valinor felt that he was too possessive of the stones since the light they captured was someone else's creation.
There were also problems between Feanor and his half-brothers. Eventually Feanor was sentenced to life outside of Tuna for a period. It was during this period that Morgoth killed the two trees and stole the Silmarils with the help of a giant spider. They also killed Feanor's father.
Feanor swore to follow Morgoth, avenge his father and recover the Silmarils. He and his seven sons took an unbreakable oath on this, calling on powers to see that they would be tormented if they abandoned their quest. Feanor's brothers agreed to follow him as well as some others like Galadriel who wanted to rule a kingdom of her own.
First they had to get to Middle Earth. Feanor demanded the ships that the elves who lived along the shore had made. They refused and Feanor took them by force, killing some of the elves who resisted.
At this point the god of death who was acting as the messenger for the rest of the Valar appeared to them and told them to return and be forgiven. If they refused they would not be allowed back. Some of them did return, do penance and were accepted back. Most did not and carried on.
And that is the original sin for the elves - leaving the Valar. Nothing they had done until then was unforgivable but when they refused to return they were abandoned and unable to return. The Valar abandoned them with the exception of the sea god who gave them what help he could.
As it turned out, they were no match for Morgoth. They thrived for several centuries while Morgoth built up his forces but eventually he overran Middle Earth except for some scattered groups.
Eventually they were redeemed by the sacrifice of Earendil. His father was human and his mother was an elf. He married Elwing who also had mixed parentage (one of her grandfathers was human and one of her great-grandmothers was a maya who took the form of a female elf). Wearing a Silmaril that Elwing's grandparents had stolen from Morgoth, Earendil penetrated the storms and currents that had kept all other mariners from reaching the Undying Lands. He went before the Valar and asked for their mercy. They granted it and overthrew Morgoth but Earendil was not allowed to return to Middle Earth. Instead he was to sail the heavens with the Silmaril showing as a star.
After Morgoth was overthrown, the Valar told the elves that all was forgiven and they could return to Valinor, he Undying Lands. Here's the second sin. Many elves refused to return. They preferred to remain in Middle Earth. They were cursed that life in Middle Earth would grow tedious and they would eventually long to return to live with the Valar.
In both cases, the elves were punished for refusing to live with the Valar. That was the sin. The Valar were God's (Iluvatar's) instruments on earth and, by refusing to live under their rule, the elves were rejecting Iluvatar.
Here's where Sauron and the rings enter the picture. The elves didn't want to leave Middle Earth but they knew their power would fade and they would eventually be forced to. Enter a disguised Sauron who had a way of capturing some's power in a ring so it would be preserved. He convinced the elf-lord Celebrimbor to help him. Celebrimbor and some assistants forged the Three rings but then they learned that Sauron would be able to rule them through the rings so they hid them until Sauron was defeated and presumed dead.
After Sauron was gone the elves used the rings to preserve their power. The three great elf-kingdoms were Lothlorien, Rivendell and the Grey Havens were all protected and maintained by the rings.
That left the elves with a terrible problem when Sauron and the One Ring reappeared. By that point the elves (and later Gandalf had been using the rings for centuries. It was established that men who wore rings of power too long were under Sauron's control, even after the rings were taken from them. That might have been true for the elves, also.
The elves had spent an age defying the wishes of the Valar that they return to the West. A sacrifice was needed to redeem them. And this sacrifice turned out to be Frodo.
There are several character arcs in LotR but the main one is the sacrifice of Frodo. He starts the story content in his life until Gandalf tells him what the ring really is. From there it's one trial after another for Frodo. The journey starts pleasantly enough with nothing worse than a marsh and tiny bugs to worry them but Frodo is wounded before they get to Rivendell and his journey after that is excruciating (in the book it takes days and Frodo's only escape from the constant pain is when he passes out).
As the journey progresses Frodo realizes that he can no longer trust his companions nor can he lead them into certain death (except Sam who refuses to leave). The RIng becomes heavier the closer they get to Mordor and it dominates Frodo's dreams. By the time they reach Mordor proper, Frodo has been wounded again and can barely drag himself along. He's given up all hope of return and only hopes to live long enough to accomplish his mission.
And then, after all that, he fails. Frodo claims the ring for himself and it is only destroyed by an accident.
Frodo's mission is accomplished and he's saved. But he wasn't really. Frodo took the sins of Middle Earth on himself and could not stay. Like the elves, there was no comfort for him in the world and he had to go with them to the Undying Lands.
And, after Frodo's sacrifice, the elves finally submitted to the Valar and returned to Valinor.