What is the narration of this story? Is it first person retrospective?

First person narratives always employ the words "I," "me," "mine," "my," etc. Edgar Allan Poe's famous story "The Cask of Amontillado" is a good example of first-person retrospective narration. "To Build a Fire" could be told in the first person, but it isn't. The narrative form is third-person with an anonymous and seemingly omniscient narrator. The narrator knows all about what is going on inside the man and even what is going on inside the dog. This anonymous narrator always refers to the man as "he," which is what makes it a third-person narrative. The narrator also knows all about the climate conditions in the Klondike and about such things as the ancestry of the dog. 

Vladimir Nabokov quotes a little rhyming couplet which he offers as a rule for first-person narration. 

The "I" in the story
Cannot die in the story.

If the "I" telling the story were to die before the end of the story, the story could not go on because the narrator would be dead. In "To Build a Fire," the protagonist dies before the story is over. The narrator tells the rest of the story from the perspective of the dog. 

Later, the dog whined loudly. And still later it crept close to the man and caught the scent of death. This made the animal bristle and back away. A little longer it delayed, howling under the stars that leaped and danced and shone brightly in the cold sky. Then it turned and trotted up the trail in the direction of the camp it knew, where were the other food-providers and fire-providers.

This is another indication that this cannot be first-person retrospective, or any other kind of first-person narration. The story is told in the third person and mainly limited to the points of view of the man and the dog, although the author Jack London interjects some straight prose exposition about such things as the setting and the weather conditions. 

In their excellent anthology titled Points of View: An Anthology of Short Stories (Revised edition 1995), the editors, James Moffett and Kenneth R. McElheny, have arranged the stories according to the narrative points of view, starting with the most subjective (e.g., "I Stand Here Ironing") and moving up to the most objective (e.g., "The Lottery"). "To Build a Fire" is not included in Moffett and McElheny's collection, but they would classify it as ANONYMOUS NARRATION--DUAL CHARACTER POINT OF VIEW. The dual characters are, of course, the man and the dog.