The obvious answer is there is no rest of the movie if he doesn't.
Most viewers think it's a huge plot hole in the story and that there is no one correct answer to explain it because the movie does not give an explanation. The movie leaves it up to the viewer to come up with an explanation, so anything that is plausible goes from that point on. Even Robert Vaughn balked at playing Chalmers initially because the story did not make much sense to him. According to him, there wasn't a story. Steve McQueen said it, too. One has to give Steve McQueen, Vaughn and the wonderful character actors in the movie much credit for almost making one forget the story and that it does not follow.
So why does Renick unlock the door? Sorry for the major spoilers. Do not read if you have not seen the movie.
Theory # 1
Renick is not very bright and he is doing what he is told to do by Ross in order to collect the rest of his money and trip to Italy. He probably was given some money up front because he had the large amount of traveler's checks and two tickets to Italy. The money is what's important to him and he figures he can retire and go to Italy with his wife. Still, one would think he would have figured out that his life was in danger somewhere along the line, with the cops protecting him and all, but to make it work we have to think Renick was dumb enough to not comprehend why he was impersonating Ross. That is, he could not figure out he was impersonating Ross so that the mob would not find real Ross and kill him. Even if Renick did not unlock the door, we see the killers kicked the door open so Stanton and Renick standing in front of the door made them both easy targets.
In the scene at the swanky Mark Hopkins hotel, Renick goes in and asks for Johnny Ross' messages and is surprised that there wasn't any. He did not realize he was being used as bait to make the organization aware that he was in San Francisco.
Theory # 2
This is more plausible, but still assumes Renick is not bright enough to figure why he is impersonating Ross. He may have thought that Ross was coming up and was going to give him his payoff. After all, he was told Chalmers was coming up and he may have been told Chalmers was in on the plot in order to protect the real Ross. Ross may have told him that he was going to be a decoy, so that until he was safe, Ross had to play the decoy. It does not totally explain why he had to take the chain off since if it was Chalmers or Ross (under protective custody), then they would be let in regardless. Still, it could be Renick is just following Ross' orders.
Theory # 3
Renick is dumb and continues to do what he is told by Ross and that was to take the chain off. It may have been Ross who hired the hit men because if the cops believe he is dead, i.e. Renick as Ross dead, then the real organization men will stop looking for him and the police would stop their investigation, too. He just wanted to ensure his men would be able to get in and commit their dastardly deed and then he would have enough time to take over Renick's identity and get out of the country with the mob money. Ross had to kill Renick's wife because she would be able to identify him.
Theory # 4
This is a variation of theory # 3. It was not Ross who hired the hit men, but the organization. Some viewers think the two hit men were seen in the opening scene and their job was to take out Renick whom they think is Ross. Of course, the professional hit men botch the job because they leave witnesses such as Stanton, the hotel clerk who could identify them, and they did they make certain Renick was dead. This became another major hole in the story.
Theory # 5
This movie was about style over substance, so it didn't matter if the story had holes. That was not the point.
Ok, so after the shooting, we got hit men from Ross or the Organization and they botched the killing of Renick who they think is Ross. Why do they have to continue to tail Bullitt after he saw them in order to find out where Renick is?
Well, we wouldn't have this cool chase scene if they do not do this. Bullitt can identify one of the hit men, so they may have decided to kill him once he realized they were following him. However, if you were going to do this, wouldn't you have your shotgun loaded much sooner?
Theory # 6
They were not tailing Bullitt to find where Renick went, but thought Bullitt "made" the shooter, i.e. he recognized one of them as the hit man at the hotel where Renick was shot, so they were out to eliminate him as a witness. This theory is full of holes because they left other witnesses. If they were going to kill a cop, wouldn't they want to do it so they could get away? Perhaps circumstances led them into the car chase, but if they were out to get Bullitt, they would have planned it better.