Manipulation planning is the problem of finding a sequence of robot configurations that involves interactions with objects in the scene, e.g., grasping and placing an object, or more general tool-use. To achieve such interactions, traditional approaches require hand-engineering of object representations and interaction constraints, which easily becomes tedious when complex objects/interactions are considered. Inspired by recent advances in 3D modeling, e.g. NeRF, we propose a method to represent objects as continuous functions upon which constraint features are defined and jointly trained. In particular, the proposed pixel-aligned representation is directly inferred from images with known camera geometry and naturally acts as a perception component in the whole manipulation pipeline, thereby enabling long-horizon planning only from visual input. Project page: https://sites.google.com/view/deep-visual-constraints