The riddle
What Has Teeth but Cannot Bite?
From The World's Best Conundrums and Riddles of All Ages, 1924
Teeth for hair
Thin, evenly spaced prongs form the teeth of a comb, and they pass through hair. Combs come in many sizes, from wide-toothed ones to fine ones with the teeth set close together. Using the mouth word "teeth" for a tool is the whole trick, and "cannot bite" is the clue that moves the listener away from animals. Everything fits, and nothing bites.
In books from 1903 and 1924
J. Gilchrist Lawson's 1924 collection, The World's Best Conundrums and Riddles of All Ages, asks the same thing in different words: "What is that which never uses its teeth for eating purposes?" The printed answer is "A comb." Dean Rivers, in his 1903 book Conundrums, Riddles and Puzzles, asks "What is a good thing to part with?" and gives the same answer, a pun on the part in a person's hair.
Other things with teeth
A comb is not the only tool with teeth and no mouth. A saw, a zipper and a gear all qualify. A saw's teeth cut wood, a zipper's teeth lock together and a gear's teeth mesh with another gear. A listener who offers one of these has found a fair answer.
Comb, and its other meanings
The word "comb" turns up in unexpected places. The fleshy crest on top of a rooster's head is called a comb, and so is the wax structure of cells that bees build, which is why that is called a honeycomb. Merriam-Webster lists all of these senses.