At an Apple store, they will ask for a student ID. I brought a friend once who went to an overseas school (was here on vacation) and they couldn't find it in their system so after a few minutes instead of researching the problem they just said "Congrats, you just went to Harvard" and gave him the discount anyway.
Online, they do not ask you for any proof what so ever. I have bought a few things with the student discount (fretting that they were going to e-mail me asking for proof or they would jack the price up but it never happened). Things just came in and all was fine.
So the answer is clearly no, you don't need to be a student. If you are caught somehow, the worst that they'll do is cancel the order and you can try again or find someone who is eligible to place the order for you. I have never heard of anyone getting into real legal trouble for trying to order via the student discount online.
Of course if this gets around and people abuse it too much, Apple may clamp down on the discount. But this is not likely. Apple likes to err on the side of making customers happy and they clearly make enough margins to cover the discount to not make it worth their while going strict on this.