late 14c., duete, "obligatory service, that which ought to be done," also "the force of that which is morally right," from Anglo-French duete, from Old French deu "due, owed," hence "proper, just" (on the notion of "that which one is bound by natural, moral, or legal obligation to do or perform"); from Vulgar Latin *debutus, from Latin debitus, past participle of debere "to owe," originally, "keep something away from someone," from de- "away" (see de-) + habere "to have" (from PIE root *ghabh- "to give or receive"). Related: Duties.
Military sense of "a requisite service" is by 1580s. The sense of "tax or fee on imports, exports, etc." is from late 14c.; hence duty-free (adv.) "free from tax or duty" (1680s), and, as a noun, "duty-free article" (1958), "duty-free shop" (by 1980).