For the compound noun teachers day, some people call it teacher day and some other call it teachers' day. However, I believe that apostrophe implies possession and so as a day cannot be possessed by a teacher but only devoted so I think it's incorrect to say teachers' day and instead teachers day should be used.
Also, is it compulsory to use capital t(T) when referring to teacher's day as:
The day usually starts with a warm Teachers Day speech to thank the teachers for their contribution. Few traditional methods include gifting beautiful Teachers Day cards and gifts to the teacher as tokens of love and respect. Some students also select popular Teachers Day quotes to convey hearty messages to their teachers and leave it at their desk.