Boethiah’s Calling Sacrifice

It could be one of two things: The follower you're using, or what you're doing when you get there.

I noticed some of the followers wouldn't do what you needed them to do. One example being one of the Dark Brotherhood initiates. I've found it works best with one of the mercenaries you can hire from one of the main cities. I used the one in Solitude. A nice list put together by the guys at SegmentNext can be found here. It shows where they are and what their names are.

When you get to the shrine with your follower, you want to talk to them and use the option "I need you to do something." Your cursor will then change, as will the action you're performing. Simply look at the shrine, and it should say "Activate sacrifical altar" or something along those lines. An animation should show your follower walking to it, and then being attached to it. From there, simply kill them by any means, the Sacrificial Dagger works, but is not required.

If you're playing on PC, you can try to install a follower overhaul, such as Ultimate Follower Overhaul (UFO) or Amazing Follower Tweaks (AFT). These should allow you to remove your follower's equipment.

It sounds to me though that your save is glitching out, something that often happens to very old, very long saves. In that case, and IF using the follower mods doesn't work (please don't try them at the same time) then you can use the save cleaner to try and clean up the game scripts on your save.

As with everything, make sure to back up your game before messing around, especially when it's in a delicate state.