Can the tensor product of two non-free abelian groups be non-zero free?

The tensor product $M\otimes N$ of abelian groups cannot be non-zero and free unless both $M$ and $N$ are free. This follows from the following facts.

1. Any subgroup of a free abelian group is free. This holds even in the infinitely generated case. Wikipedia has a proof in its article on free abelian groups.
2. Under certain conditions on $M$ and $N$, for non-zero $n\in N$ the homomorphism $$ \begin{align} &M\to M\otimes N,\\ &m\mapsto m\otimes n, \end{align} $$ is injective. It is easy to see that this holds when $N$ is free and $n$ is a basis element. It also holds when $M$ and $N$ are both torsion-free. You could prove this by extension of scalars to reduce it to the case of vector spaces over the rationals (and vector spaces are always free). This does not hold if you merely assume that $N$ is torsion-free, as pointed out by Jack Schmidt in a comment (and I must apologize for making the mistake of assuming this in my original answer).
3. Tensor products are right exact. That is, if $0\to A\to B\to C\to 0$ is an exact sequence of abelian groups, then $$ M\otimes A\to M\otimes B\to M\otimes C\to 0 $$ is exact. In particular, if $M\otimes A$ maps to zero, this shows that $M\otimes B$ and $M\otimes C$ are isomorphic, which I will use to quotient out the torsion subgroups before applying 2.

Now, suppose that $M\otimes N$ is non-zero and free. Let $T$ be the torsion subgroup of $N$, so that $N/T$ is torsion-free and $0\to T\to N\to N/T\to 0$ is an exact sequence. By 3, $$ M\otimes T\to M\otimes N\to M\otimes(N/T)\to 0 $$ is exact. However, as $T$ is torsion, $M\otimes T$ will also be torsion, so its image in the torsion-free group $M\otimes N$ is zero. This gives an isomorphism $M\otimes N\cong M\otimes(N/T)$. Applying the same argument with the torsion subgroup $S$ of $M$ shows that $(M/S)\otimes(N/T)$ is isomorphic to the free group $M\otimes N$. Then, as $M/S$, $N/T$ are torsion-free, picking any non-zero $n\in N/T$ and using 2 gives an injection $$ \begin{align} &M/S\to (M/S)\otimes(N/T),\\ & m\mapsto m\otimes n. \end{align} $$ So $M/S$ is isomorphic to a subgroup of the free group $(M/S)\otimes(N/T)$ which, as stated in 1, means that $M/S$ is free. Similarly, $N/T$ is free. Applying 2 again, the homomorphism $$ \begin{align} &M\to M\otimes(N/T),\\ & m\mapsto m\otimes n \end{align} $$ is injective for any basis element $n\in N/T$. So $M$ is isomorphic to a subgroup of the free abelian group $M\otimes(N/T)\cong M\otimes N$ and, using 1 once more, must be free. Similarly, $N$ is free.

Any example must arise from infinitely generated groups. Suppose $A$ and $B$ are finitely generated, non-zero, non-free, abelian groups. Being non-free means there is torsion (since we're in the finitely generated case). If every element of $B$ has finite order, then every element of the tensor product has finite order. So, assume $b \in B$ has infinite order, and let $a \neq 0$ be an element of order $n$. Then $a \otimes b$ is non-zero and has order dividing $n$. Hence, $A \otimes B$ has torsion and, consequently, it is non-free.