What does a zero tensor product imply?

Try the contrapositive. If $Ann(M) + Ann(N)$ is a proper ideal, what interesting kind of ideal can you choose that contains it? Try using that ideal to help.

Also, as a general rule, to show that $M\otimes N$ is non-zero, try to find a map to some quotient that is simpler to understand (and so simpler to show is non-zero).

Here is an alternative proof, using the facts

1. $\mathrm{supp}(M \otimes N) = \mathrm{supp}(M) \cap \mathrm{supp}(N)$ (Hint: Nakayama and Linear algebra)
2. $\mathrm{supp}(M) = V(\mathrm{Ann}(M))$

This easily implies $V(\mathrm{Ann}(M \otimes N)) = V(\mathrm{Ann}(M) + \mathrm{Ann}(N))$. Hence, $M \otimes N=0$ iff the set is empty iff $\mathrm{Ann}(M) + \mathrm{Ann}(N)=A$.