From the UESP wiki's "Skyrim Puzzle Totems - Bugs" article:

A common bug is that the rings on the dragon claw doors may not rotate, or the door may not open after the correct combination is set. Possible fixes are:

• Save and then immediately load that save.
• Load either the auto-save from the entrance, or a save from before you entered.
• Exit the location, save, and load that save.
• "Reset" the door by selecting the right combination and try the claw, then turn each ring once and try the claw again. Do this until the correct combination re-appears and try again.
• (On Xbox 360) Clear the hard drive cache.

From the UESP wiki's "Skyrim Puzzle Totems - Bugs" article:

A common bug is that the rings on the dragon claw doors may not rotate, or the door may not open after the correct combination is set. Possible fixes are:

• Save and then immediately load that save.
• Load either the auto-save from the entrance, or a save from before you entered.
• Exit the location, save, and load that save.
• "Reset" the door by selecting the right combination and try the claw, then turn each ring once and try the claw again. Do this until the correct combination re-appears and try again.
• (On PC) Type tcl into the Console to clip through the door. Reenter the code once past. Note that unlock will not work.
• (On Xbox 360) Clear the hard drive cache.