An integrated numerical model that describes the operation of a renewable-energy-based system for a building’s heating, cooling, and domestic hot water needs is described in this study. The examined energy system includes a vapor compression multi-source heat pump, PVT collectors, borehole thermal energy storage, and water tanks. Energy balance equations for the collectors and the tanks are coupled with correlations for the heat pump and the piping losses within a thermal network approach. The non-linear system of equations that arises is solved by employing in-house software developed in Python v. 3.7.3. The performance of the numerical tool is validated against measurements collected during the pilot operation of such a system installed in Athens (Greece) for two 5-day periods (summer and winter). It is shown that the proposed model can predict, both qualitatively and quantitatively, the building’s energy system performance, whereas limited deviations from the experimental findings are mostly observed when highly transient phenomena occur. The numerical tool is designed with flexibility in mind and can be easily adapted to accommodate additional energy-system configurations and operational modes. Thus, it can be utilized as a supporting decision tool for new energy systems’ designs and the optimization of existing ones.