I've watched several times this video animation of Neil's death in Tenet. At the 45 second mark of the video, we see inverted Neil approaching the locked door. Now, from Neil's inverted perspective, Ives and The Protagonist (TP) are already on the other side of the door, contending with Volkov.
Why, then, does Neil need to intervene by unlocking the door for Ives and TP? Couldn't Neil see that Ives and TP already made it through the door (he sees them reverse-fighting Volkov)?
Inverted Neil needs to close the door, so that they are opened from Protagonist's perspective. Neil realises that protagonist and Ives won't be able to open the door, so he know he must help them ("I’m the only one who could’ve got that door open in time, right Ives?"). Tenet's universe seems to obey some sort of Novikov self-consistency principle (or as Neil puts it: "What’s happened’s happened."). So even when you realize what happened thare is really no point in trying to change it, you will not succeed. For example Protagonist learned this after his 'failed' inverted car chase.
From reverse Neil's point of view, all he knows is they manage to get in. But he doesn't know how or if he was involved. The best way of making sure, is for him to be on the other side of the gate to potentially open it for them.
The two answers posted above, along with a third viewing of the movie, has led me to an acceptable conclusion. Considering the movie's dialogue shown below:
Ives to Protagonist: "How'd you get that lock open?"
Protagonist to Ives: "Wasn't me."
This establishes that it was neither Ives nor Protagonist that opened the door.
Neil to Ives: "I'm the only one who could've got that door open in time, right, Ives?"
Ives to Neil: "Well, I don't know any locksmiths as good as you."
This further establishes that Neil was the only one that could have possibly opened the door.
So, I'll post this answer to my question: Neil must have been the one that opened the door because the other two character absolutely couldn't. However, Christopher Nolan preserved cause and effect throughout the entire movie except for the scene where Neil opens the door (hence, the confusion of so many viewers of this spectacular movie).
