Personification is the literary technique of assigning human characteristics to inanimate objects. In "The Monkey's Paw," the paw itself is not seen to be human-like, but to be somehow animated. Because of this, the scene where the paw moves is not an example of personification; in fact, Mr. White make a different comparison:
"It moved," he cried[...] "As I wished it twisted in my hands like a snake."
This explicit comparison does not evoke feelings of humanity, or cause the reader to assign human characteristics to the paw itself. However, the following scene, where Mr. White watches the fire, is a good example of personification.
He sat alone in the darkness, gazing at the dying fire, and seeing faces in it. The last face was so horrible and so simian that he gazed at it in amazement.
(Jacobs, "The Monkey's Paw," gaslight.mtroyal.ca)
Seeing images in random movement is pareidolia, and is present in everything from clouds to TV static. The human mind likes to assign meaning to things, regardless of the actual form or shape; fire is in a constant state of motion, and so it is easy to see images in it. Mr. White has been put into the mindset of the supernatural and the scary, and so when he sees faces in the fire, his mind assigns them human characteristics appropriate to his mental state. Monkeys have generally human-like faces, and so when Mr. White sees a "horrible simian" face in the fire, it is an example of personifying the fire to coincide with the story themes.