The story
The noun is derived from fine (“particularly slender”) + tooth (“sharp projection”) + comb. The verb is derived from the noun.
Etymology from Wiktionary, CC BY-SA 4.0.
fine-tooth comb
Today it means: A comb with fine, closely spaced teeth, especially one used for removing head lice and their nits (eggs) from the hair; a nit comb. Full definition →