This is not true in general.
The main problem is that tensor product can create torsion and co-torsion and reflexive modules have neither. (The dual of a finitely generated module is reflexive, that is, isomorphic to its own double dual).
Even if you assume that $M_1$ and $N_1$ are both reflexive over an integral domain, $M_1\otimes N_1$ may have torsion and this gives a counterexample since reflexive modules are duals. Let $M=M_1^\vee$ and $N=N_1^\vee$. Then $M^\vee\otimes N^\vee=M_1\otimes N_1$ has torsion while $(M\otimes N)^\vee$ being a dual does not.
Notice that in this case the map $$M^\vee\otimes N^\vee \to (M\otimes N)^\vee$$ is not injective. An interesting feature of this example is that $M$ and $N$ are torsion-free.