Stereotypes harm society by spreading lies and making people less human. Stereotypes, even those with an element of accuracy, are inaccurate lies because no group of people is completely the same. They make people less human because we tend to perceive people as their stereotype rather than as a "regular person." Consciously we may strive to treat people without prejudice but entrenched stereotypes often drive our actual behavior.
An example of the damage from a "positive stereotype": the employer of an Asian employee assigns him to accounting tasks because he believes the stereotype that all Asians are good at math. If that employee is not good at math his actual best talents will not be used and the employer will not have the best person doing the job.