Assuming Stanley's $P$-partition conjecture holds, the regular Schur labeled skew shape posets with underlying set $\{1,2,\ldots, n\}$ are precisely the posets $P$ such that the $P$-partition generating function is symmetric and the set of linear extensions of $P$, denoted $Σ_L(P)$, is a left weak Bruhat interval in the symmetric group $\mathfrak{S}_n$. We describe the permutations in $Σ_L(P)$ in terms of reading words of standard Young tableaux when $P$ is a regular Schur labeled skew shape poset, and classify $Σ_L(P)$'s up to descent-preserving isomorphism as $P$ ranges over regular Schur labeled skew shape posets. The results obtained are then applied to classify the $0$-Hecke modules $\mathsf{M}_P$ associated with regular Schur labeled skew shape posets $P$ up to isomorphism. Then we characterize regular Schur labeled skew shape posets as the posets whose linear extensions form a dual plactic-closed subset of $\mathfrak{S}_n$. Using this characterization, we construct distinguished filtrations of $\mathsf{M}_P$ with respect to the Schur basis when $P$ is a regular Schur labeled skew shape poset. Further issues concerned with the classification and decomposition of the $0$-Hecke modules $\mathsf{M}_P$ are also discussed.