No. The American Cancer Society recommends it for women 21 and older, or 3 years after starting sexual activity, whichever comes first. However the American College of Obstetricians and Gynecologists recommends it only for women 21 and over, regardless of sexual history. This is because pap changes in adolescent women usually go away by themselves and the additional tests and treatment for abnormal pap tests can do more harm than good in this age group. Even after they do start doing pap tests, the newer recommendations are to do them every 2-3 years in most women, not every year.
Doing tests unnecessarily CAN cause harm. False positive tests lead to more invasive procedures which in some cases can cause permanent damage. There is no question that the pap test is a useful screening test for cervical cancer, but if overused it can cause problems. The chance of getting cervical cancer as an adolescent is very, very small, and pap tests do not test for the kind of cancers that are made worse by estrogen (breast cancer and endometrial cancer which are also unheard of in adolescents).