There is a theorem that said if $X$ is a continuous poset, then if $x \ll y$, there exists $z\in X$ such that $x\ll z\ll y$, where $\ll$ is the way below relation.
I am not sure why the continuity is needed. Can somebody point me to a example of a non-continuous poset without an interpolation property?
Here is one such example.
You can check that this poset has the interpolation property, but it is not continuous since $\{x : x \ll \omega \} = \{ 0 \}$.
Edit: the question was actually on non-continuous posets without the interpolation property (see BeerR's comment below). To get such a poset you can keep the example above and paste a copy of the real line below the element denoted $0$. This prevents $0$ from being compact. Hence you come up with $0 \ll \omega$ with no element $z$ such that $0 \ll z \ll \omega$.