As the title says, given a general $q$-binomial $\binom{n}{k}_q$, is there some general result regarding its value at a root of unity, $q = \exp(2\pi i r/N)$?

So, the canonical answer is the q-Lucas theorem, as pointed out in the comments.

This was proved in
*Olive, Gloria*, **Generalized powers**, Am. Math. Mon. 72, 619-627 (1965). ZBL0215.07003.

Enumerative Combinatorics, second ed.) and equate coefficients of $x^k$ to get an answer. A special case is Exercise 1.98. $\endgroup$Generalized powers, Amer. Math. Monthly 72 (1965), 619–625, equation (1.2.4). (The proof, which is not difficult, is apparently in her 1963 Ph.D. thesis, which I have not seen.) The theorem has been rediscovered many times since then, but I have not come across any earlier occurrence of it. $\endgroup$4more comments