The first way that suspense is created, particularly by Sergeant Morris, is that he only tells a part of the story. He informs his audience that the first man who used his three wishes with the monkey's paw wished at last for death, implying that something must have gone terribly wrong. But he gives no details and the reader has no insight into the story thanks to the position of the narrator. So the reader is left to create the details or possibilities on their own, helping to lend the story a significant amount of suspense.
Mr. White too adds more suspense by leaving so much up to the reader to discern, not giving clear voice to his thoughts and allowing the reader to figure out the rest. There is also a moment in the chess game where Mr. White makes dangerous and ill-considered moves, foreshadowing the poorly thought out wish that starts the main action of the story in motion.