A: Most likely, yes. In general, an employer has the ability to dictate reasonable appearance standards for it employees—so long as those standards don’t illegally discriminate based on an individual’s race, gender, religion, etc. However, several states have regulations limiting an employer’s ability to proscribe appearance standards. So, it is always prudent to consult with your employment attorney regarding legal matters—especially concerning sensitive matters such as appearance.
As with all employee-related matters, it is safest to relate school policy to the needs of the job. In this case, stating that “no tattoos may be visible” can be related to the faculty member’s role as a role-model (in appearance, decorum, temperament, etc.) to students. Such a requirement is best spelled-out in a clear “Appearance Policy” published in the employee handbook.
The flip side of the issue, however, is, “Is the school unnecessarily limiting its pool of candidates by proscribing job ‘requirements’ that aren’t truly requirements but are rather preferences?” If the tattoo issue isn’t central to the school’s conception of “role modelness” for faculty members, the school might be best staying neutral (i.e., saying or enforcing nothing with regard to tattoos).