Ok, so I'm trying to compile knowledge about all these dragon claws you receive for various dungeons and puzzle doors.
First of all, in case you are in here because you can't handle the first simple puzzle, examine the claws in your items menu if you want to know what combo opens a door with them. Anyway here are the claws I have found and their statuses as quest items:
Gold Claw: Pretty much everyone gets this, and after clearing the first dungeon with it, it's no longer a quest item, but it can be given to the shopkeeper in Riverwood to make him happy. Ivory Claw: Gotten in a dungeon while doing the followup quest to assemble the amulet split among the three brothers (one in Gold Claw dungeon, one in Ivory, one in Emerald). Check the Writ of Sealing or something like that on the first boss by the Eye of Magus to start the followup quest. Doesn't lose quest item status until the end of this followup quest when the amulet is reforged in a fourth dungeon. Emerald Claw Same as above, but found in a different dungeon. Also note: The Ivory and Emerald claws are reused in the final (fourth) dungeon in the quest, the gold is not.
Ruby Claw Found in a dungeon during the first quest for the Bard's college. Used to solve a door puzzle in that dungeon but does not lose quest status afterwards. If anyone has any information on further use for this item, I'd love to know how to finally drop it.
I have head of several more claws existing, notably Sapphire, Coral, Iron, Diamond, and Ebony Claws If you have any information on these claws, what quests they are needed for, any combinational quests that might get my Ruby Claw to lose quest item status, or any claws currently unlisted, please post here.
I got the sapphire claw from doing a quest in the town at the bottom of the mountain, ya know the town at the start of the path to High Hrothgar.
Well anyway, go to the tavern and listen for the girl to start talking about a ghost near the barrow near town to the barkeep. After she's done, talk to the barkeep. He'll tell ya how the place is haunted and some guy went there two years ago and never came back. Volunteer to go and investigate, bring back the journal and give it to the barkeep. He'll give ya the sapphire claw, now ya can take the claw back to the barrow ya just came from and go deeper inside!
I think saphires the one I have. I had to clear the Barrow, find the guy who's pretending to be a ghost, then go back to the store to report and I was given it. Then I had to go all the way back down to use it.
Didn't seem worth the hassle, I don't recall getting any decent loot from that Barrow. --- Ignorance is bliss