Panin and Yagunov on a form of Poincaré duality for theories represented by a T-spectrum.
Alex Shannon mentioned some kind of very general notion of Poincare duality I think, maybe involving infty-cats.
The Beilinson conjectures should maybe be thought of as a kind of PD (over F1???).
http://mathoverflow.net/questions/9457/do-chains-and-cochains-know-the-same-thing-about-the-manifold
Can Langlands duality be thought of as Poincare duality in some sense? Both types of duality are reflected in functional equations.
Notes from the Heidelberg-Strasbourg seminar 1966/1967, in Friedman scans folder. Contains for example notes from an exposé of Verdier.
nLab page on Poincare duality