Allan / Slushie,
It should be possible to ammend the registration confirmation email function such that the emails are always sent to the forum admin, rather than the applicant.
That way, you'd learn of every membership application and could click on the email's link to activate it, or go into the admin section and delete the new (inactive) account, then email the applicant to tell them "sorry".
If you ammended the terms and conditions page to include a note that non-student applications would be declined, that should put off most bogus applicants to minimise your workload; You could also alter the post-application page to say something like "Thanks for your application, it's being considered".
I suppose it depends how many genuine students you have - if it's hundreds, you'd have an impossible amount of work to do, but if it's only a couple of dozen each year you should be OK.
I must admit, I haven't done any digging for this specific reason, but I believe it would be possible; however -boRg-'s suggestion seems a lot simpler! (and each year you could batch-delete members who hadn't posted in the last 12 months, ie, registrants who weren't students and were thus left in the "no posting allowed" starting group)
Just a couple of penneth...