The essential task of integrated electricity-heat systems (IEHSs) is to provide customers with reliable electric and heating services. From the perspective of customers, it is reasonable to analyze the reliabilities of IEHSs based on the ability to provide energy services with a reasonable assurance of continuity and quality, which are termed as service-based reliabilities. Due to the thermal inertia existing in IEHSs, the heating service performances can present slow dynamic characteristics, which has a great impact on the service satisfaction of customers. The neglect of such thermal dynamics will bring about inaccurate service-based reliability measurement, which can lead to the inefficient dispatch decisions of system operators. Therefore, it is necessary to provide a tool which can analyze the service-based reliabilities of IEHSs considering the impacts of thermal dynamics. This paper firstly models the energy service performance of IEHSs in contingency states. Specifically, the nodal energy supplies are obtained from the optimal power and heat flow model under both variable hydraulic and thermal conditions, in which the transmission-side thermal dynamics are formulated. On this basis, the energy service performances for customers are further determined with the formulation of demand-side thermal dynamics. Moreover, a service-based reliability analysis framework for the IEHSs is proposed utilizing the time-sequential Monte Carlo simulation (TSMCS) technique with the embedded decomposition algorithm. Furthermore, the indices for quantifying service-based reliabilities are defined based on the traditional reliability indices, where dynamic service performances and service satisfactions of customers are both considered. Numerical simulations are carried out with a test system to validate the effectiveness of the proposed framework.