In my opinion, the story totally implies that Herbert's death is dues to fate and not to chance.  But it is never actually said explicitly.

First of all, it seems likely that it is fate.  That seems to be the whole point of the story -- that the paw is bringing an evil fate to anyone who is greedy enough to make a wish.

Second, we see a bit of foreshadowing when the old man makes his wish.  Herbert says that he does not see any money.  And then he goes on to say "and I bet I never shall."  This sort of implies as well that fate is going to prevent him from ever seeing the money.

So there is no proof that fate is driving events, but it sure is implied.

Was Herbert's death by fate or chance in "The Monkey's Paw"? What are some examples from the story to prove this?

I'm not so sure that the story was meant to be a horror story, primarily. Certainly a horrible thing happens in the story, the mangling to death of poor young Howard, but the story is just as philosophical as it is a tale of horror.

After all, much of the horror is in the mind of the reader, for no conclusion can be drawn as to why Howard died. And did Howard's disfigured, bloody corpse claw its way out of the grave, trudge all the way to his parents' home and knock on there door in the dark of night because of a wish? Or was it but the wind and your imagination?

Chance or fate? Who's to say? And how can one hope to prove one's theory one way or the other?

A wish was made. A death occurred. Money was obtained. There was a late-night knocking at the door. So?