When the story begins, we might assume that we are getting a peek at the thoughts of the unnamed man, who finds the day not just "cold and gray" but "exceedingly cold and gray" (1); but it is unclear whether these are his thoughts or the narrator's. The narrator describes the incredibly cold temperatures and the man's frozen face without much emotion or investment. It's really a toss-up to decide who cares less about the poor guy—the narrator or the wolf dog.
There's really only one time when the narrator seems surprised, and that's when an exclamation point is inserted after the falling snow from the tree has made sure that the man's fire is "blotted out!" (23). This could just reflect the man's shock, though, and not the narrator's concern. It might also be meant to shock us (and boy does it work). For the most part, though the narrator seems not to care one way or the other whether or not this guy dies, and wishes only to comment in a factual (sometimes downright preachy) way on what might have contributed to the his death.