Tensor product of abelian group with itself is zero

What about $A=\Bbb Q/\Bbb Z$? $A$ is both torsion and divisible, so $(1/m)\otimes (1/n)=n(1/nm)\otimes 1/n=(1/nm)\otimes (n/n)=0$.

ADDED IN EDIT

On the category of Abelian groups, Tor is left-exact in both arguments, so if $B$ is a subgroup of $A$ then $\text{Tor}(B,B)$ is a subgroup of $\text{Tor}(A,A)$. If $A$ has nontrivial torsion, then $A$ has a subgroup isomorphic to $C_n$, a cyclic group of finite order $n\ge 2$. Therefore $\text{Tor}(A,A)$ has a subgroup isomorphic to $\text{Tor}(C_n,C_n) \cong C_n$, and so is nonzero.