Hair dye does not kill lice. Dye has no effect on lice at any stage of their life cycle. There is some anecdotal evidence that indicates that bleaching one's hair kills adult lice, but it does not seem to have any effect on the eggs.
The key to removing lice effectively involves using a proven product to eliminate the lice, such as a lice-killing shampoo, and combing the hair with a fine-toothed louse comb to remove the eggs, nits and lice. This treatment must be repeated within a week to 10 days to ensure that any newly hatched lice are also killed.