I don't know a very general answer. Your duality on cell complexes resembles Verdier duality and has a local nature, but some of these equivalences aren't like that. E.g. $K = (0 < 1 < 2)$ and $K^{op} = (2 < 1 < 0)$ are isomorphic as posets and so we get $D(K) = D(K^{op})$ from that, but I am pretty sure $K$ doesn't have a Verdier dualizing sheaf.

But here is an answer to a narrower question: when is there an equivalence $D(K) = D(K^{op})$ that is similar to the one on regular cell complexes? I mean an equivalence that is Verdier-like in the sense that it takes the standard injectives on $K$ to the skyscrapers on $K^{op}$, up to a shift.

For each $x \in K$, define $J_x:K \to \mathrm{Mod}_k$ by
$$
J_x(w) = \begin{array}{cc}
k & \text{if $w \leq x$}\\\
0 & \text{otherwise}
\end{array}
$$
These are the indecomposable injective objects in the abelian category of functors $K \to \mathrm{Mod}_k$.

The $J_x$ have a simple Hom pattern: $\mathrm{Hom}(J_x,J_y) = k$ if $x \geq y$. All other Homs (and Exts) vanish. (They form an "exceptional collection" in $D(K)$.)

The simple objects are the skyscrapers $\delta_x$. You can compute the Homs and Exts between $\delta_x$ and $\delta_y$ by writing down an injective resolution of $\delta_y$ whose $p$th term is
$$
\bigoplus_{y = y_0 \geq y_1 \geq y_2 \cdots \geq y_p} J_{y_p}
$$
The differentials have degree $+1$. Then $\mathrm{Hom}(\delta_x,\text{that injective resolution})$ is a complex whose $p$th term is
$$
\bigoplus_{y = y_0 \geq y_1 \geq y_2 \cdots \geq y_p = x} k
$$
This is the cochain complex that computes something like the relative cohomology $H^*(N,\partial N)$ where $N$ is the nerve of the interval
$
\{p \in K \mid x \leq p \leq y\}
$
and $\partial N$ is the subcomplex of simplices that don't contain the edge $[x < y]$. So a necessary condition is for this $\partial N$ to be a homology sphere.

I think this condition is sufficient too, except you have to worry a little bit about the dimensions of those spheres. You have to be able to choose integers $d(x)$ for each $x \in K$ so that the dimension of that homology sphere is $d(y) - d(x)$ or something.

Somebody once told me that this condition on the intervals in posets has a standard name, maybe "Gorenstein star posets" but I am not sure I am remembering that right.