Okay. I proved that $\operatorname{Ann}(N_1) + \operatorname{Ann}(N_2) \subseteq \operatorname{Ann}(N_1 \cap N_2)$, where $N_1$ and $N_1$ are submodules in some module. I am now trying to show by example that the other inclusion doesn't necessarily hold. I tried considering $\Bbb{Z}_8$ as a $\Bbb{Z}$-module and looking at $N_1 = (2)$ and $N_2 = (4)$, but this didn't work. I could use a hint as to where I could find an example.
Also, in the context of ring theory, what does $(2)\Bbb{Z}_8$ denote?
Take $k$ to be a field and look at $R=k[x,y]/(x^2,xy)$. Then $\mathrm{Ann}((x))=(x, y)$ and $\mathrm{Ann}((y))=(x)$, but $(x)\cap(y)$ is the zero ideal in this ring, so it has annihilator $R$.
$\mathbb Z_8$ typically denotes $\mathbb Z/8\mathbb Z$.
The reason ideals in $\mathbb Z_8$ won't work is because it is a commutative Ikeda-Nakayama ring, and that means that the annihilator equation you gave will hold for ideals in that ring.