I'm interested in proving the following claim:
Lemma: Let $M$ be an aspherical, closed 3-manifold which is not hyperbolic. Then, there exists a finite cover $M'$ of $M$ and an incompressible embedded torus $T$ in $M'$, such that $T$ is non-nullhomologous in $M'$.
I'm aware that if $M$ is Seifert fibered, then there exists a finite cover $M'$ of $M$ such that $M'$ is a circle bundle over a surface. I would like to say that we can take a further finite cover $M''$ which is a circle bundle over a surface $F$, where $F$ contains an embedded non-separating curve $\gamma$. Then the torus $T$ in $M''$ corresponding to the curve $\gamma$ is non-separating and thus non-nullhomologous.
If the above is correct, then it suffices to prove the Lemma in the case where the JSJ-decomposition of $M$ does not contain any Seifert fibered pieces. However, here I am stuck.
Yes, this is true but the proof is somewhat long. You can extract it from the proof of the main result in
Hempel, John, Residual finiteness for 3-manifolds, Combinatorial group theory and topology, Sel. Pap. Conf., Alta/Utah 1984, Ann. Math. Stud. 111, 379-396 (1987). ZBL0772.57002.
Note that the paper operates under the "Haken" assumption, but it is only to ensure that Thurston's Geometrization Conjecture applies to the class of manifolds under consideration. Now, it's not a conjecture but a theorem (due to Perelman).