$\Omega(n)$ is the number of prime divisors of $n$, counted with multiplicity.

For $z=-1$, $z^{\Omega(n)} = \lambda(n)$ is the Liouville function, and it's known that $\sum_{n\leq x}\lambda(n) = O(n^{\frac12 + \varepsilon})$ is equivalent to the Riemann hypothesis.

Is this the case for all roots of unity other than 1? Is this a known result?

If we look at the Dirichlet series $F_z(s) = \sum_{n=1}^{\infty} \frac{z^{\Omega(n)}}{n^s} = \prod_p {\frac{1}{1 - z p^{-s}}}$, we have for the product of roots of unity of order $n$: $\prod_{k=1}^n F_{e^{\frac kn2\pi i}}(s) = \prod_p {\frac{1}{1 - p^{-n s}}} = \zeta(n s)$, and using Mobius inversion we can look only at the primitive roots of unity of order $n$, $\prod_{k\leq n\\ (k,n)=1} F_{e^{\frac kn2\pi i}}(s) = \prod_{d | n} \zeta(d s)^{\mu(\frac nd)}$, but I'm not sure how to continue from here and treat each of the roots separately.

Heuristically, $\Omega(n)$ is normal with increasing mean and variance, so for any order of a root of unity $k$ I'd expect $\Omega(n) \mod k$ to be uniform as $n\to \infty$, so $\sum z^{\Omega(n)}$ should be approximately a random walk in 2D and thus $O(n^{\frac12 + \varepsilon})$.