We introduce a novel decision-diagram-based framework for generating intersection cuts to strengthen polyhedral relaxations of bounded mixed-integer nonlinear programs (MINLPs) with general nonlinear constraints. Building on decision diagrams (DDs) that provide outer approximations of MINLP feasible regions, we propose the concept of DD-free sets, which are convex regions whose interiors do not intersect the sets represented by the DDs. By exploiting the underlying graphical structure, we develop efficient methods for constructing these regions without requiring restrictive assumptions on the functional form of nonlinear constraints or access to distance oracles. We investigate the maximality properties of DD-free sets and devise algorithms for obtaining maximal forbidden regions within several structured classes. The resulting intersection cuts exploit geometric information retained in DD representations, enabling further strengthening of relaxations beyond what can be achieved through conventional DD-based convexification. We also introduce interval DDs to facilitate the representation of general nonlinear constraints and establish convergence results for intersections of constraint-specific DDs, providing a theoretical foundation for their integration into global optimization frameworks. Computational experiments on challenging open instances from MINLPLib, encompassing a broad range of nonlinear and nonconvex structures, demonstrate that the proposed approach can substantially improve dual bounds and achieve greater optimality-gap closure than state-of-the-art global optimization solvers. These findings highlight the potential of exploiting graphical representations of infeasible regions to develop stronger cutting planes for general MINLPs.