In this paper, we investigate the sectional contravariant curvature of a doubly warped product manifold ( fB×bF,g˜,Π=ΠB+ΠF) equipped with a product Poisson structure Π, using warping functions and sectional curvatures of its factor manifolds (B,g˜B,ΠB) and (F,g˜F,ΠF). Qualar and null sectional contravariant curvatures of ( fB×bF,g˜,Π) are also given. As an example, we construct a four-dimensional Lorentzian doubly warped product Poisson manifold where qualar and sectional curvatures are obtained.