It is a measure of the sheer folly of the unnamed protagonist in “To Build a Fire” that even the dog he takes along with him on his expedition knows that it is nothing more than a suicide mission. At various points in the story, the dog demonstrates by its behavior and body language that neither he nor his master should really be out in such treacherous conditions.
In that sense, the dog is indeed the man's foil. His innate sense of danger serves to highlight the arrogance, foolishness, and ignorance of a man who thinks that he can not just defy nature, but defeat it. The dog clearly understands nature and respects it. This is the exact opposite of the man, who is ultimately undone by his ignorance and lack of respect for a world of which he forms an intrinsic part.
The dog instinctively adjusts its behavior to the rhythms of nature, which is more than can be said of its master—who, far from adjusting to nature, is actively trying to impose himself upon it. It says a lot about this hapless individual that a mere canine has considerably more wisdom than himself. But then that's because the dog, unlike the man, has an instinctive realization that it is a humble animal with all the limitations that that entails.