If $0$ is the zero-object $ \Longrightarrow F(0) $ is the zero object when $F$ additive
Let $0_M$ and ${\rm id}_M$ denote the zero map and identity map on an $A$-module $M$. We have
$$M=0\iff 0_M={\rm id}_M.$$
Since $F$ is a functor, $F({\rm id}_M)={\rm id}_{FM}$. Since it's also additive, $F(0_M)=0_{FM}$.