Author W. W. Jacobs never tells the reader why Sergeant-Major Morris turned pale while discussing his third wish in "The Monkey's Paw." The result of his third wish could not have been a happy one, however.

    "Well, why don't you have three, sir?" said Herbert White, cleverly.
    The soldier regarded him in the way that middle age is wont to regard presumptuous youth. "I have," he said, quietly, and his blotchy face whitened.
    "And did you really have the three wishes granted?" asked Mrs. White.
    "I did," said the sergeant-major, and his glass tapped against his strong teeth.

Later, the Whites used their own third wish to erase the aftermath of their second wish. Mr. White "frantically breathed" as he made it, and Mrs. White gave "a long loud wail of disappointment and misery" when she felt only "a cold wind" and saw nothing but a "deserted road" outside.

In this story, Morris does not make any wishes.  And Mr. White does not grow pale after he makes his third wish.

Morris does grow pale (his face whitens) during the story.  It happens when Herbert Morris asks him why he does not have three wishes himself.  He says that he already has, and his face grows pale.

This implies that his three wishes turned out really badly, just as White's wishes will.  They turned out so badly that it makes him get pale just thinking about how bad it was.