What role does humor play in the narration of A Christmas Carol?
I think it is absolutely typical of the style of Dickens that he manages to combine humour in his narration with the serious moral that shines through the text. From the start, the narration provides an ironic commentary on Scrooge and his failings, ever-keen to point out what kind of character that he is in a humorous fashion. Notice how he is described:
The cold within him froze his old features, nipped his pointed nose, shrivelled his cheek, stiffened his gait; made his eyes red, his thin lips blue; and spoke out shrewdly in his grating voice. A frosty rime was on his head, and on his eyebrows, and his wiry chin. He carried his own low temperature always about with him; he iced his office in the dog-days; and didn't thaw it one degree at Christmas.
Note here how hyperbole is combined with matter-of-fact description for humorous effect, presenting Scrooge in his chilly manner by literally referring to how his presence spreads the cold ice that is so much a centre of his character. And in this fashion the novel continues, mixing humour with its narrative and the serious moral message that Dickens has for us and for Scrooge.