What exactly causes Paul's death?
In terms of the story's symbolism and themes about greed, Paul dies because his mother can not be satisfied. Even if he had continued winning more and more money, she would have simply craved more and more. Therefore, in lieu of his mother's insatiable greed, there would have been one of two outcomes: Paul's death or an end to his lucky streak.
If we're talking about what literally kills Paul, that is more difficult to determine and is subject to interpretation. This story is designed like a fable or a fairy tale, so the precise cause of Paul's death is not specified and it may not be essential to the meaning of the story. But hypothetically, the cause could be related to stress. Paul has taken it upon himself to become the breadwinner and the "man of the house." He is only a child but has taken on the Freudian burden of replacing his father and becoming like a husband to his mother. The mental and physical strain of his "rides" combined with the psychological (probably unconscious in Paul) confusion of his roles in relation to his mother perhaps caused enough stress to lower Paul's immune system to the point that he dies of some subsequent ailment.
In the end, Bassett says, "But, poor devil, poor devil, he's best gone out of a life where he rides his rocking-horse to find a winner." In other words, Paul is better off dead. The strain was too much for him. It was selfish and thoughtless for Paul's mother and uncle, even Bassett, to allow Paul to take on this burden. Paul works himself to death. This fable, in the modern context, is about man's (or "a boy's") futile attempt to substitute money for love. Again, this is as impossible as satisfying an insatiable consumer like his mother. He dies because he works himself to death trying his best to complete an impossible or never-ending task.