Let $R$ be an integral domain and let $X$ be a torsion module over $R$. Then $Tor_n(X,Y)$ is a torsion module for every $n≥0$.
Solution 1:
Take a projective resolution of $Y$. Then, tensoring with $X$ yields only torsion modules and thus the subquotients (in particular, the homology groups of the complex, aka the various $\mathrm{Tor}$ modules) are still torsion.