You have blank lines between your equations there. Blank lines can't appear in math mode and maths environments and flalign is no different.
You should either get rid of them:
\begin{flalign}
rez_1[\;] &= \lim_{s \to 0} \left[\frac{2(s + 2)(s + 4)}{(s + 1)^{2}(s^{2} +
2s + 2)}\frac{1}{z - e^{sh}}\right] = \frac{8}{z - 1} \\
rez_2[\;] &= \frac{1}{1!} \lim_{s \to -1} \left\{\frac{d}{ds} \left[\frac{2(s
+ 2)(s + 4)}{s(s^{2} + 2s + 2)}\frac{1}{z - e^{sh}} \right] \right\} =
\frac{-14z + (14 - 6h)e^{-h}}{(z - e^{-h})^{2}} \\
rez_3[\;] &= \lim_{s \to (-1 + j)} \left[\frac{2(s + 2)(s + 4)}{s(s + 1)(s + 1
+ j)} \frac{1}{z - e^{sh}}\right] = \frac{3 + j}{z - e^{(-1 + j)h}}
\end{flalign}
Or comment them out:
\begin{flalign}
rez_1[\;] &= \lim_{s \to 0} \left[\frac{2(s + 2)(s + 4)}{(s + 1)^{2}(s^{2} +
2s + 2)}\frac{1}{z - e^{sh}}\right] = \frac{8}{z - 1} \\
%
%
%
rez_2[\;] &= \frac{1}{1!} \lim_{s \to -1} \left\{\frac{d}{ds} \left[\frac{2(s
+ 2)(s + 4)}{s(s^{2} + 2s + 2)}\frac{1}{z - e^{sh}} \right] \right\} =
\frac{-14z + (14 - 6h)e^{-h}}{(z - e^{-h})^{2}} \\
%
%
%
rez_3[\;] &= \lim_{s \to (-1 + j)} \left[\frac{2(s + 2)(s + 4)}{s(s + 1)(s + 1
+ j)} \frac{1}{z - e^{sh}}\right] = \frac{3 + j}{z - e^{(-1 + j)h}}
\end{flalign}

Off-topic but I'm unfamiliar with rez. As you have it, it is typeset as the product of the variable r, the variable e and the variables z1, z2 etc. If it's supposed to be a single unit \mathit{rez}_{1} or similar might be appropriate.
%on the otherwise blank lines. – Teepeemm Oct 22 '19 at 17:47