December 2023 reference for Open That Door! being stuck:
First I dropped the quest, left the area, reloaded UI and it didnt fix it.
Complete the nearby quest marked as "Objectives" Monument Maintenance - which requires you to get two pieces of a crumbling statue for repairing the middle statue. There is a piece located both on the north and south stone ramps going up to the vault door. These pieces you have to carry by hand and cannot use a flying, ground, or dragonriding mount.
Monument Maintenance ITEMS:
/way 58.63, 72.48 Arm and Weapon
/way 62.72, 72.43 Head
/way 62.35, 69.47 Orb
Complete the nearby quest marked as "World Quests" Data Gathering - this quest requires you to use a scanning beam which shows up as a special item similar to other quest use items on the bottom middle of your screen. You have to scan the nearby yellow/passive creatures. You may have to fly around a bit to find them, you cannot scan the same one twice. These seem to have a 1/5 chance of becoming aggressive when you scan.
After you complete these, the quest should continue but you may need to drop the quest and restart it or reload UI.
Side note:
This may be a bit late but for anyone playing catchup, as of December 2023:
This quest was stuck for me, Rannan wouldnt move and the quest wouldnt continue. A previous comment mentioned some of this info but I want to be more specific.