The increasing penetration of renewable energy and the growing frequency and severity of extreme weather events pose a significant challenge to the secure and economic operation of local energy systems. This paper presents a risk-averse scheduling model of the integrated electricity and heat system (IEHS) for resilience enhancement against contingencies. A tri-level formulation is proposed to formulate the worst line outage events and identify the optimal resilience enhancement strategies. The main idea is to find the least-cost scheduling solution that meets the predefined resilience requirements. The model incorporates multi-energy network operations that integrate network reconfiguration in the PDN and mass flow rate control in the DHN. This approach provides more flexibility for load restoration in the face of uncertainties. A nonlinear thermal flow model is adopted to realistically capture thermal dynamics, and a tractable linear approximation is then proposed to balance accuracy and tractability. To solve this tri-level mixed integer program, an efficient solution method based on the nested column-and-constraint-generation algorithm is developed for the proposed IEHS scheduling model. Numerical studies indicate that the proposed model provides optimal solutions that are cost-effective to enhance the resilience of IEHS at minimum expense.