+0
Which is correct?

Happy Teacher's / Teachers' / Teachers Day.
+0
It is Happy Mother's Day / Happy Father's Day, because children have one mother, one father.

Although students may have several teachers, one for each subject, if they are conveying this individually to each of their teachers, use the singular form:

Happy Teacher's Day

If there is a banner put up by the students of the school, wishing all the teachers at the school a happy day, then use the plural form:

Happy Teachers' Day
Comments  
Teachers: We supply a list of EFL job vacancies
Happy Teacher's Day
Teachers'