No. For example, let $T'$ be your favorite complete theory, and let $T$ be the empty theory in the same language (with no axioms). $T$ is incomplete.
But there is a trivial interpretation of $T$ in $T'$: take the identity translation from the language of $T$ to the language of $T'$. Every theorem of $T$ (which are just the first-order validities) translates to a theorem of $T'$.
In general, the operation of "forgetting axioms" is an interpretation (where the theory with fewer axioms is interpreted in the theory with more axioms).
Maybe I should add that there is a notion of an "essentially undecidable theory". This is a theory $T$ such that for any theory $T'$ interpreting $T$, the set of consequences of $T'$ is undecidable.
In particular, $T$ is incomplete, and if $T'$ is a computably enumerable theory interpreting $T$, then $T'$ is incomplete (since the set of consequences of a complete computably enumerable theory is decidable).
The classic example of an essentially undecidable theory is Robinson's Q.