Community integrated energy system (CIES) couples multiple energy forms and links, it is important to cope with the inadequacy of flexibility for realizing efficient utilization and reliable supply. Based on the detailed modeling of central energy station, district heating network (DHN) and building loads, a unified scheduling model is established by fully considering the coupling characteristics of the sources, the networks, and the loads. Meanwhile, the flexibilities from adjustable capacity of thermal storage device and the indoor temperature on the demand side, as well as the energy storage of DHN, are utilized to improve the economy. Furthermore, a model predictive control (MPC) based robust scheduling strategy is proposed to maintain and utilize CIES flexibility for enhancing the uncertainty adaptability. The scheduling framework consists of rolling optimization and robust constraint generation. The rolling optimization schedules the system with a better adaptation to flexibility demands, and the robust constraints generate adjustable margin for building demands by modifying the upper/lower and ramping limits. After the linearization of nonlinear terms in the model, case studies are conducted based on the data of a typical day in summer. The results show that the unified source-network-load scheduling strategy can give full play to the flexibilities of different energy links with a better operation economy. Additionally, the MPC-based optimization can well adapt to the rolling forecasting uncertainties, and the schedule generated by advance reserving means has stronger robustness for real-time errors.