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.

The answer by anonymous mathematician seems essentially correct to me.

One thing to remember is that (in all math depts. I'm aware of, including Harvard's, and mine) grad admissions are reviewed by a faculty committee, who have nothing to do with imposing/collecting application fees and so on --- so there is certainly no correlation between department rules/guidelines and things like fees.

The committee members will do their best to assess candidates based on all the information they have available, using their best judgement as to how well the candidates will perform in their program. At Harvard in particular, they have their pick of the very strongest candidates from all over the world. GRE's are just one small component of their evaluation process, but are useful --- if someone doesn't do very well on the GRE, this is a flag. Essentially all the students admitted by top programs such as Harvard will have gotten close to a perfect score on the GRE subject test, though, and so the GRE serves more as a negative indicator against admission (if a candidate didn't do we'll) than as a positive indicator that someone should be admitted.

But of course occasionally someone who is unquestionably very strong will have done poorly on the GRE, or not done it at all, and the department will want to have the option of being able to waive it as any kind of formal criterion. Hence the slightly ambiguous wording on their website.

(One thing to remember is the once the dept. committee makes its selection, there is a bureaucratic process within the university where the list of admitted candidates has to be given to/accepted by other university entities, such as the university's graduate school. The department probably has to give some explanation/justification for its selection, and some discussion of the criteria it used. The GRE is presumably one of these, but they don't want to put themselves in a straitjacket of not being able to ignore it in special cases, and hence --- as Anonymous Mathematician notes --- they have a slightly ambiguous, flexible, policy.)

This is not a grey area. This is simply a contradiction. The first paragraph you've shared says it is required and the second sure seems to imply that they are only strongly recommended.

Since it's easier to use the single word "required" accidentally and sloppily than it is to write that second paragraph by accident, my guess would be that they mean that the subject GREs are only strongly recommended, not required.

Let's be very clear though: The opinion of people here doesn't matter. The specific departmental admissions officers and committee are the only opinions that matter since they will make the decision. If you see contradictory information on their website, point it out to them and ask for a clarification. They will clarify the issue for you and probably even fix the page.

Finally, even if something is "only" very strongly recommended, it's still strongly recommended! Top programs are extremely competitive and it's up to you to do everything you can to strengthen your application. You might be the next generation's superstar but unless you've already done incredible, it can be extremely hard to tell this. Fantastic scores on the subject GRE can help make your case. Not doing it at all might suggest some insecurity on your part that you might not do well. Take the test unless you have an extremely good reason not to.