Even though I have the correct answer to the puzzle, it won't open. At first the rings didn't even move, so I exited and reentered the temple. That worked, but now the door just won't open with the correct ring positions. I don't know how I'm supposed to beat the game! I did try exiting the entire place and reentering again, but nothing changed. I ended up just reloading my game to Dragonsreach and I did some other quests.
One possible workaround described on UESP:
Sometimes the dragon claw door may not rotate, or the door may not open. Simply save and then immediately load that save to fix.
If that doesn't work, try exiting the temple, save and load from the outside. Go inside, check if both gates on the first puzzle is open, they shouldn't be (only the one you need). Proceed towards the third puzzle where you "reset" it by selecting three dragons and inserting the key, then select the right combination.
If you are playing on PC, you could use the console command
TCL to simply walk through the door. Use the console command again to resume normal interactions with floor and walls.
TCL doesn't work if you have an item selected in the console. Right click or double left click and item to clear your selection (the item number of what you have selected will disappear, allowing you to no-clip freely).
I had this same problem. I backtracked until I got a loading screen, then went back to the door. It opened. Saving/quitting/loading had previously not worked.
At first, I had the problem that none of the circles moved at all, then I saved the game and load it right away and the circles started to turn.
I put the right combination and nothing... Saved, loaded... leaved the room, saved, turned the Xbox off, tuned on, returned to the circle and still nothing... I kept trying randomly till I set the right combination again AND as soon as I pressed the diamond claw, I went to the middle ring (mouth/butterfly) and pressed (trying to change the animal), while the claw was in the animation. Worked for me.
In my game it did the same thing but no matter what I did the door would not open. But I did find a way to get past having to go through the temple altogether thus skipping the diamond door and all that straight to the top.
This way is very complicated, first you have to go up the mountain near where you arrived and travel along the top of the mountain sometimes sliding down a little and you can get to a point where you can jump to the top platform. This took me a while to do just so you know