No, there is no useful research analog of FizzBuzz.
FizzBuzz is a rather limited test of programming ability. An incompetent candidate may fail at it, but passing this test in no way qualifies someone to work as a programmer. Some programming jobs attract lots of incompetent applicants, and it's worth filtering them out, but FizzBuzz is just a small step in that direction. The only reason we've heard of it is because it makes a great story when someone applies for a job but can't pass the simplest test.
For faculty hiring, there's no need for such a crude filter. The number of highly qualified candidates is much greater than the number of openings, and filtering out the incompetent is not a big deal. (If you spend a lot of time worrying about whether your candidates are incompetent, then something has gone catastrophically wrong with the search process.) Instead, the hard part is deciding who the very best candidates are, and FizzBuzz-style questions won't help with that.
For graduate student admissions, competence testing is more relevant. However, there's a fundamental difference from hiring a programmer. In that case, you are looking for someone who already has programming experience, and FizzBuzz helps filter out people who are bluffing about how experienced they are. Someone who fails could still learn to be a fine programmer, but you don't care since you're looking for someone who already knows how. On the other hand, most applicants to graduate school have only limited experience with research, and that's OK. You're looking for potential, rather than experience, and that's trickier to evaluate. I'm skeptical that any simple, clear-cut test can reliably predict research potential.