I would like to get into a top university for graduate school, specifically in computer science. Would taking the non-required mathematics subject based GRE test make my application more competitive? Since it's not required, would the people doing admissions even consider it as a data point?
No one here can speak for some admissions committee. They might look at it, might not. If they have strict rules they might even need to ignore it. It won't do you any harm, though and there is an outside chance it will make some small difference. But focus more on the things that they do ask for.
And, for what it's worth, the competition at top schools is fierce. Get good letters of recommendation.
While there exists a small chance that an excellent result on a non-required GRE Subject will increase your chances to get admitted into a graduate school, I am not convinced it is worth the time and money.
The time to prepare and take this take could be more efficiently spent into preparing a better statement of intent, getting good letters of recommendation, or polishing your CV. I would consider doing some research/teaching-related volunteering as a better way to take a chance to slightly improve your graduate school application, which might be reflected in all three aforementioned documents.
Totally agree with @Buffy's answer: it's hard to say about a particular admissions committee; however, the expected impact of you having this result is very small. And the graduate school applications have a lot of items you can focus on anyway, and those items really make the difference.
Many things that impress one committee member will not impress others. I suspect it will not help much, and possibly could hurt your chances a little in some cases. What is most important are: letters of recommendation, academic records and experience.
Many people going into top CS programs are strong in math, so it doesn't really help to set you apart too much (except see 2), which is what you need to do to get in to the really competitive programs. Even for math programs, your score on the GRE is not really considered much. (The math on the GREs is not that advanced--it's mostly used as a way to weed out people who definitely shouldn't be going to top math programs.) To impress committees with your mathematical abilities, you should take some advanced math classes, or at least math-heavy theoretical CS classes, and do really well in them. Or you can do really well in the Putnam or similar.
Second, people might wonder why you've take the math GRE and sent this info to them. The first thing people will probably guess is that you are also considering going to grad school in Math. For many people, this won't be a concern, but maybe if you're on the borderline, depending on how the rest of your application reads, some committee members may worry that CS isn't your first choice.
Disclaimer: I'm in math, not CS, so it's possible my impression is mistaken, but if so, hopefully someone will correct me.
Would taking the non-required mathematics subject based GRE test make my application more competitive?
It doesn't matter whether it will or not. In my opinion, you're fetishizing these tests.
Would scoring well on a non-required GRE Mathematics Subject Test make me more competitive?
Well, IMHO you're already too focused on competition. I respectfully suggest you focus more on what kind of actual research you're interested in, rather than your "score" against that of others, or the "score" of your future university against that of others etc.
If you had a potential subject you could contact relevant research groups directly.
If you're not sure what you want to pursue, scientifically, also consider the possibility that it might not be right for you to go into a graduate program right now (at least not one which is research-focused).
Many people - myself included - have made the mistake that graduate school is simply the next phase of undergraduate studies, with research being a sort of tougher homework; if this is ringing familiar to you, then - you're in for a rude awakening a couple of years down the line. A lot of graduate students experience anxiety and depression, often because of going in with the wrong motivation. Maybe this won't happen to you, but still - take your time thinking about these questions before you head off to some grad school.