If a person has already eaten one Devil Fruit, what will happen to him if he eats another Devil Fruit?
Their body would explode without a trace, and they would die...
During the Enies Lobby arc, in chapter 385, we see how Jabra gets scared when seeing the Giraffe Fruit and the Bubble Fruit. He claims that a rumor exists that says that when an ability user comes close to a Devil Fruit, the Devil will come out of the fruit and start fighting with the Devil residing within his body and that would make his body explode. This was immediately countered by Bluenote saying that Grandline scientists have already uncovered that the body would only explode if two fruits are eaten by the same user. Therefore it is believed and apparently proven that eating two Devil Fruits would make a person's body explode.
Later during the Marineford arc, in chapter 577, we have seen an exception to this case. Blackbeard was seen to have two Devil Fruits at the same time. When Whitebeard died, he supposedly extracted the Devil from Whitebeard's body and put it into his own. How Blackbeard was able to extract it and whether his body did not explode due to it being different, due to the Dark Fruit powers or due to a different reason, can be read here.