The centerpiece and longest part of The Silmarillion is the "Quenta Silmarillion," or "History of the Silmarils." To say it's dry would be accurate. Much of it reads like the begats in the Bible. Which is appropriate, since it's essentially a history of the Elves from their creation to the end of their dominion on Earth.
Why, then, is it not called "Quenta Quendi" or something like that?
Instead, it's called the History of the Silmarils, those three stones made by one of the first elves, Fëanor, into which he put the light and power of the Two Trees, i.e., of the most mysticalest of the mystical ecstatic godnature of the Tolkien equivalent of the Garden of Eden, or something like that. See, Morgoth (Sauron's daddy) steals the Silmarils, and Fëanor and his sons swear to get them back, and never let anyone else have them. This oath basically guides the rest of Elven history, leading most of the Elves out of Eden and into the world, and at the same time introducing murder into Elvenkind; it's no exaggeration to say that Fëanor's sons' obsession with this oath leads to the utter downfall of the Elves.
Obviously, the Silmarils are connected with Tolkien's version of Original Sin. They get you kicked out of the Garden. But what is it about the Silmarils - it's not like the apple in Genesis, where it's put right in front of you and you're told not to eat it (Tolkien saves this motif for the men of Numenor). It's not a temptation.
No, the Elves' original sin is making things. Fëanor makes the Silmarils, and they're wonderful - even the Valar (the angels of Eden) love them. But once the light of the Trees has gone from being a ubiquitous phenomenon to a (mostly) self-contained thing that can be possessed, well, then it's all over but the weeping. If you can possess it, then somebody else can want to possess it, and kill you for it.
In short, I think the Elves' original sin is objects. Things. This is never quite spelled out, but it's all there in the Silmarils. The Elves are immortal, and they have what seems to readers an infinite capacity for apprehending beauty. They should be content just to be. But they aren't. They set about making things, the Silmarils. Making things is beneath Elves, or should be. Later we get an echo of the Silmaril making in the figure of Eöl, who has learned smithying from the dwarves, and who becomes a sort of foreboding, unElvish figure. But really, he's not that much different from Fëanor.
Sauron with his rings is another echo, and this is how Silmarillion can help make sense of LOTR. When Sauron made his rings he imbued them with his power. This made them powerful tools, but of course it made him hugely vulnerable.
You make a thing thinking it's going to be an increase for you, but instead it diminishes you.
Materialism. That's the big sin in Tolkien, isn't it?