If having the monkey's paw results in death, why didn't Herbert die? What was the bad thing that happened to him? He had it before the sergeant did.
The monkey's paw does result in a bad fate for those who use it. Remember a curse was put on the paw by a fakir (a holy man) for, "He wanted to show that fate ruled people's lives, and that those who interfered with it did so to their sorrow." That doesn't mean, though, that the person who makes the wishes is going to die.
The first man who used it did die. However, that man used his third wish to ask for his death. This implies that his first two wishes must have resulted is something so terrible that he couldn't bear to live anymore. This foreshadows what will happen to the Whites.
It is never stated what the sergeant-major's three wishes were. But we can infer from the opening scene that they have not resulted in anything positive. He seems like a tired or haunted man. This too is why he urges Mr. White to let the paw burn, or if he must use it, to wish for something small and reasonable.
Remember too that Herbert doesn't use the monkey's paw. His father does. Herbert just tells him what to wish for. Mr. White wishes for 200 pounds. Keeping with the lesson the fakir wanted to teach those who use it, Mr. White gets his money, but it comes at the expense of his son's life.
Part of the lesson might be too that there are worse things than death. The sergeant-major seems a broken man. Mr. White, who uses his last wish to wish his son away again, and his wife will always feel the loss of their son from their lives.