Energy management is crucial for various activities in the energy sector, such as effective exploitation of energy resources, reliability in supply, energy conservation, and integrated energy systems. In this context, several machine learning and deep learning models have been developed during the last decades focusing on energy demand and renewable energy source (RES) production forecasting. However, most forecasting models are trained using batch learning, ingesting all data to build a model in a static fashion. The main drawback of models trained offline is that they tend to mis-calibrate after launch. In this study, we propose a novel, integrated online (or incremental) learning framework that recognizes the dynamic nature of learning environments in energy-related time-series forecasting problems. The proposed paradigm is applied to the problem of energy forecasting, resulting in the construction of models that dynamically adapt to new patterns of streaming data. The evaluation process is realized using a real use case consisting of an energy demand and a RES production forecasting problem. Experimental results indicate that online learning models outperform offline learning models by 8.6% in the case of energy demand and by 11.9% in the case of RES forecasting in terms of mean absolute error (MAE), highlighting the benefits of incremental learning.