If there are untoward allegations regarding a teacher/student relationship, you simply do not demand access to FB to prove such allegations.
You lay the burden of proof on the accused/suspected to defend themselves. If they are unable to prove their innocence, perhaps by remaining adamant that their FB account remain hidden from school authorities, then you fire them. Explain to them what will happen when they refuse to defend themselves against such allegations, and the repercussions of such, and fire them.
boom. done.
Except that's not how due process works.
You're not allowed to go on a fishing expedition to find something to charge a person with. You have to have the evidence before hand.
Otherwise, the government/employer could just accuse anyone of anything and get full access to whatever the hell the want. Clear 4th ammendment violation, and clear wrongful termination lawsuit.
If they have the picture and it's innocuous, there is no reason to "assume" that there are any worse pictures on the facebook page. If you don't have the picture in the first place, there's no reason to believe there's any wrongdoing.
BUT OH THINK OF THE FUCKING CHILDREN! GOD FORBID THEIR TEACHERS/TAS ENGAGE IN SEX OR DRINK ALCOHOL! HOLY FUCKING SHIT, PEOPLE IN AUTHORITY DOING SOMETHING YOU BELIEVE IS WRONG!!?!?!?!?!! KILL THEM WITH FIRE!
You don't like the possibility that your kid's teacher might, you know, have a normal life outside of their job? Send your kid to fucking Catholic school.
Fucking hell, our society is fucking useless. This nannystate, nosey bullshit has to stop. Whatever happened to a reasonable expectation of privacy?