Interpretation of "GRE subject test is required, but your application will be considered anyway"
This is certainly poorly and confusingly worded, but I think I can explain the reasoning based on similar issues I've seen with other departments.  Here's my interpretation:

The department expects every applicant to take the GRE subject test.  However, they know that every year a small fraction of the applicants screw up and don't manage to register in time to take it in the fall.  It's impossible to take the test at the last minute, so at that point there's no way to fix the situation.  The department doesn't want to automatically reject these applicants, for exactly the reason you mention (what if someone's the next Terry Tao?), and they are worried that if they list the GRE as an absolute requirement, then the university administration might force them to do that.  On the other hand, they also don't intend to offer applicants the discretion to decide for themselves whether taking the subject test would strengthen their applications.  They want everyone to take it, with the possibility of forgiveness for outstanding candidates who screw up.  These goals get combined into the confusing statement that it's mandatory but your application can still be considered without it.

I don't know how the Harvard math department evaluates applicants with no GRE scores, but I can tell you how my department does.  We're a little happier if we see an apologetic note about how the applicant intended to take the test but failed to do so in time, but of course we know such a note proves nothing.  Our default assumption is that the applicant was worried about doing poorly, so we start asking questions like "How badly would we guess they might have done?" and "Would we have accepted them anyway, even with low GRE scores?"  If the application is otherwise fantastic, then the chances of acceptance remain good.  However, these are the cases in which we would have cheerfully made excuses for low scores (maybe the applicant just isn't good at standardized tests or had a bad day), so nothing is gained by skipping the exam.  An otherwise marginal applicant with no GRE subject test score will be rejected, and a strong but not fantastic applicant will be at a disadvantage.

So my best guess as to what the policy means in practice is something like "if you don't take the exam, we'll be annoyed at your irresponsibility and we'll evaluate your application with a presumption that you wouldn't have done well on the exam, but we'll still consider admitting you."  This means you have a strong incentive to take the exam if you can.


  On one hand, it seems possible that they would want to "consider" applications without the GRE subject test, in that they receive the admission fee and that they can discard the application with very little time and effort 


In a typical U.S. department, application requirements and graduate admissions decisions are handled pretty much entirely separately from application fees.  For example, many people on the committee won't even know offhand what the current application fee is or where the money goes within the university.  In particular, whoever wrote the policy about GRE scores probably wasn't thinking about application fees at all.