This depends on whether you're asking what the purpose of the paw is in the story, or what the purpose of the story is in itself.
According to the sergeant, the purpose of the Paw is to educate those who would "interfere with fate" - although the Paw's manner of education, according to the sergeant, is dangerous and abhorrent. The Paw is important to the story because it initiates the action; it makes dreams "attainable", but also twists them, and stirs up greed in the Whites that causes them to overlook the very clear signs (throwing the Paw on the fire) that this object is more trouble than it's worth. The Paw itself isn't necessarily more important than the Whites in this regard, and in the end it's a mix of the two that can be said to be responsible; the Whites for making wishes, and the Paw for distorting them.
The Paw also has purposes outside of the story; for example, this was written in 1902, when India was still part of the British Empire, but also "exotic". By virtue of its strange and dangerous properties, yet tied to a seemingly wise and beneficial lesson from a "holy man", the Paw may represent foreign customs and powers, and the dangers that they may hide even when they seem intended to be good.
The most direct purpose of the Paw is simply to illustrate that humans cannot control events as much as they may think they can. The fact that the Whites had their wishes twisted is not so much malice on the Paw's part, but representative of the fact that we cannot see all the outcomes of our actions. Had the Whites specifically stated that no one should die in order for their 200 pounds to be received, the Paw may have simply dropped a meteor on their house instead, eliminating the benefit of paying off the mortgage.
The story starts with Mr. and Mrs. White, who are visited by Sergeant-Major. He tells them about his trip to India and the Monkey's Paw. The paw apparently gives them 3 wishes of whatever they'd like. The first wish they made was for 200 pounds.
The next morning, their son Herbert went to work. They later were visited by a stranger telling them their son had been injured at work and did not make it. They received a compensation of 200 pounds (they got their wish).
After this incident, they wished Herbert to be alive again. He comes back as a zombie. Their third wish fixed everything, and all was quite.
The purpose of the story is, as I interpret it, to be careful what you wish for. Because of the White's greediness for their wishes and money, they got it in an unexpected way, and realized they no longer wanted it. It;s human nature to wish for things to better yourself, but you must think of others as well.