The coordinated operation of different energy systems, such as electrical, gas, and heating, can improve the efficiency of the whole energy system and facilitate the larger penetration of renewable energy resources in the electricity generation portfolio. However, appropriate models considering various technical constraints of the energy carriers (e.g., gas system pressure limit and heat losses in the district heating networks) are needed to effectively assess the true impact of integrated energy system (IES) operation on the overall system's performance. This article proposes a flexible unit commitment (UC) problem for coordinated operation of electricity, natural gas, and district heating networks, called multicarrier network-constrained unit commitment (MNUC), to minimize the operation cost of the IES. Besides, an integrated demand response (IDR) program is considered as a promising solution to improve consumers' electricity, gas, and heat consumption patterns and to increase the power dispatch of combined heat and power units. Multienergy storage systems are also included in the proposed model to decrease the impact of multienergy network constraints on the overall system's performance. To model the uncertainties involved in the operation of the three networks, a combined robust/stochastic approach is preferred in the MNUC problem considering multicarrier energy storage systems and the IDR program. Numerical results show that the whole operation cost of the IES has decreased by 2.58% considering the IDR program and multienergy storage systems.