I don't think there is any logic to this. It's just for dramatic effect. The climax of the chapter is the "accusation from beyond the grave". If the revived man had lived on, West would presumably have had to kill him to stop him going to the police and this wouldn't have made a good ending to the story.
I suppose it's possible the man (Robert Leavitt) really might have had a weak heart and he suffered a heart attack due to the stress of revival. I've always assumed that West just made up the "weak heart" as part of the story to deceive his assistant, but I suppose West could have decided to kill Leavitt when he discovered Leavitt had a a weak heart. The reason would be that if Leavitt's death ever came to light everyone would assume it was a heart attack rather than murder.