No you don't.
My sister is a teacher and she bought from Apple store under education pricing. They did not ask for any proof (comms was through her personal mail, not .edu.sg email address).
If you read the T&Cs, Apple basically reserve the right to ask you for proof you are qualify. If you cannot show proof, they will charge the difference to your credit card. So, you can always try. If you so heng to kena check, too bad.
BTW, not just for poly & uni. ITE and JC also can. Not just lecturers also. As long as staff qualifies liao (such as office clerk). Staff (but not student) also from any type of school (Kindergarten, Pri & Sec schools) also eligible. MOE staff also eligible.