The operator learning has received significant attention in recent years, with the aim of learning a mapping between function spaces. Prior works have proposed deep neural networks (DNNs) for learning such a mapping, enabling the learning of solution operators of partial differential equations (PDEs). However, these works still struggle to learn dynamics that obeys the laws of physics. This paper proposes Energy-consistent Neural Operators (ENOs), a general framework for learning solution operators of PDEs that follows the energy conservation or dissipation law from observed solution trajectories. We introduce a novel penalty function inspired by the energy-based theory of physics for training, in which the functional derivative is calculated making full use of automatic differentiation, allowing one to bias the outputs of the DNN-based solution operators to obey appropriate energetic behavior without explicit PDEs. Experiments on multiple systems show that ENO outperforms existing DNN models in predicting solutions from data, especially in super-resolution settings.