Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic $p > 0$. Is there an affine Lefschetz theorem and Poincaré duality for sheaves represented by finite flat commutative group schemes of $p$-power order for flat cohomology of $X$?
For a surface and certain sheaves, I have found Milne's "Duality in the flat cohomology of a surface".
No. Roughly speaking, finite group schemes are objects with slope between 0 and 1 and so there is no Poincare duality without twisting except on curves (see the article of Artin and Milne for that). There is a flat analogue of the etale $\mathbb{Z}_l(r)$ duality --- see the article of Milne you mention and later articles. I think the affine Lefschetz theorem fails already for $\alpha_p$.
