I'm not going to go back to comment, which would necro some very old threads on this bug:

https://forums.elderscrollsonline.com/en/discussion/566442/vvardenfell-heart-of-a-telvanni-quest-bug (2021)
https://forums.elderscrollsonline.com/en/discussion/591566/the-heart-of-telvanni-bugged (2021)
https://forums.elderscrollsonline.com/en/discussion/582288/the-vvardenfall-quest-the-heart-of-a-telvanni-is-broken (2023)

There are probably older ones, but those turned up pretty quickly in a forum search and should be sufficient support for the proposition that this problem continues to exist. I hit it last night (bug report duly submitted, no response as of the time of writing) with the exact same problem (door to the final section of the mine cannot be interacted with, so the quest cannot progress to completion). Quitting and coming back did not fix it, relogging did not fix it, abandoning and reacquiring (five times thus far) did not fix it, making different choices in the dialogues (lie/truth and knockout/kill and then trying both on both choices) had zero effect. My initial impression was that it could not be reacquired, but I finally remembered that it starts in Tel Aruhn rather than Sadrith Mora and picked it up again just fine.

There is apparently some dialogue between two NPCs which is supposed to happen before they progress through earlier doors, but this did not happen. Worth noting is that the two NPCs were waiting just outside the mine entrance after I had frustrated on that last door for a while and went out the way I came. They had dialogue that should have happened before their initial entry, but that didn't progress anything, either.

How about a nice 10th Anniversary present of fixing this? There is an update due to go live in a couple of weeks, so there is even a timely opportunity.

EDIT: managed to get it to complete. (1) Quit the game while stuck at that last door. (2) waited - about 12 hours, but I was done for the night anyway. (3) Logged back in, which put me outside the mine and reset everything inside. (4) NPCs did their dialogues at every door and the quest progressed as it should have. Side observation: kill/knockout choice doesn't seem to matter. I'd already started that run with kill and had no knockout option available in the first room, but I did wait on the NPCs to run ahead of me before doing the guard captain and they waited at the door while I did that. For the second room, I took care of the guards near the door and then followed the NPCs to the next door where they did their little chat, which made the door interactable. Quest completed normally from that point onward. Best guess is that quitting the game and waiting for a bit before coming back in was what it took to get everything to click into place.