Let X be a Banach space and $M \subset X'$ a subspace of its dual space.
If X is reflexive we know the following statements are equivalent:
(i) $M$ is dense in $X'$
(ii) $x \in X$ and $\phi(x)=0$ for all $\phi \in M$ implies $x=0$
Now (ii) $\rightarrow$ (i) is not true if X is non-reflexive.
I struggle however to come up with a counterexample.