So I was asking the Graybeards about words of power and they sent me to Valthume. I get to the room with the iron claw that sits on a small pillar with a pressure plate, which then tiggers a fire trap when you pick it up. There are three spiders in the room too. Problem is when I try and turn/activate the inner, middle and outer ring on the door to line up the right symbols by pressing the 'A' button on my controller, it doesn't do anything. I've tried reloading different saves, but end up having the same problem.
I got this exact same thing to happen. Here's what I did to get around it (This may be specific to me, but maybe this'll help out at least someone):
Note: This is on the 360.
After getting to the room and having the rings not do anything when I activate them, I left dungeon, headed back to whiterun and did some stuff there. After going back, the rings would rotate for me, but the answer didn't unlock the door so I was still softlocked.
Note: When I first entered the Valthume dungeon. I was playing on a friends 360 and he didn't have the game installed.
I had saved right in front the door and went back to my 360 to see if that would magically fix it. It didn't. I then reloaded to a save on my 360 (which already had the game installed, btw) where I hadn't got the quest to get the word of power in Valthume yet. I then went to Valthume, ignoring the greybeards, and redid the dungeon. I got through the Iron Claw door just fine.
Now, there's a ton of variables that I had changed here, but my guess would be having to do something with whether the game is installed or not. Make sure to go to a save where you hadn't entered the dungeon yet, because I remember hearing somewhere that any area you first discover will lock itself to your level at that moment (which may lock in the bug if the game isn't installed). Again, I haven't done a full test on what exactly caused the bug, but I hope this can help out at least someone who's OCD about getting all of the shouts.
Note: In case you don't know, to install the game on your 360 go to the dashboard and highlight the game icon. Press the 'X' button and then highlight install game and press 'A'.
tl;dr: Make sure the game is installed before entering the dungeon so the iron claw door won't bug out on you.