The atmosphere of the story is entirely horror; there is a brief scene of familial comfort and home life at the beginning, and then events move forward steadily, without pausing, and descriptions become negatively representational instead of neutral or positive. Mr. White:
...gaz[es] at the dying fire, and see[s] faces in it. The last face was so horrible and so simian...
These descriptions at first are subtle; while the narrative seeks to show things as normal, the language becomes more and more negative, with short sentences and sharp exclamations from the characters.
"I'm sorry--" began the visitor.
"Is he hurt?" demanded the mother.
The visitor bowed in assent.
These language choices, as well as the shortness of the narration, give the story a sense of claustrophobia, that there is little room for the characters or even the reader to relax and breathe. Simple descriptive language allows the final part to be the most terrifying:
There was another knock, and another. The old woman with a sudden wrench broke free... He heard the chain rattle back and the bottom bolt drawn slowly and stiffly from the socket. Then the old woman's voice, strained and panting.
(Jacobs, "The Monkey's Paw," gaslight.mtroyal.ca)
Each part adds to the sense of unease: the first part is warm, with fire imagery, the second part is neutral, with both fire and death, and the third part is cold and harsh, with guttering candles and dark passageways. These elements give the story a morbid atmosphere, and the sense that normalcy in the universe has been damaged.