In "The Monkey's Paw," what role does fate play in the lives of the White family?
Fate, as spelled out in the text, controls the lives of all people. This is a deterministic viewpoint, the idea that every action is pre-determined and that all people are simply living out their assigned, fated roles. This view does not necessarily require a higher being, but usually does require some sort of spiritual rulebook that cannot be broken without magic. The monkey's paw itself serves that function; it is a tool which can break determinism and cause an unfated event to happen, albeit with negative consequences.
"It had a spell put on it by an old fakir," said the sergeant-major, "a very holy man. He wanted to show that fate ruled people's lives, and that those who interfered with it did so to their sorrow. He put a spell on it so that three separate men could each have three wishes from it."
(Jacobs, "The Monkey's Paw," gaslight.mtroyal.ca)
The White family is the last to be affected by the paw; Herbert White dies to provide an insurance payment to his parents, allowing them to pay off the house. When Mrs. White asks for Herbert to be returned to life, it is not specified that he be healed of his injuries; Mr. White intuits that he will be a horribly mangled monster, not their son. He is unable to change his wife's mind, and so Herbert returns, and Mr. White is forced to use up the last wish to remove Herbert. This shows that, once determinism is broken, fate steps in to return their lives to the equilibrium; the money is received, and it is not acceptable to have the money and Herbert together. Since they were not meant to receive this extra money, something of equal or greater value must be removed.