Interval constrained multi-objective optimization problems (ICMOPs) have at least one objective function or constraint containing interval uncertainty parameters, which is popular in practical engineering applications. There are very few algorithms for solving these problems, and practical constrained optimization problems usually have discontinuous feasible domains. For the ICMOPs with discontinuous feasible domains, a feasibility rule based on intervals is presented by defining the interval-constraint violation degree. Based on this, an interval constrained violation degree guided interval constrained multi-objective optimization algorithm is proposed. This algorithm takes the decomposition-based interval multi-objective evolutionary algorithm as the framework. Firstly, Latin hypercube sampling is utilized to explore feasible domains in the search space, and multiple evenly distributed sampling points constitute an initial population. Then, the reference vector is periodically adaptively adjusted based on the individual’s interval constraint violation degree or interval crowding distance. Finally, the double difference mutation operator is employed to generate new individuals, and the neighborhood individuals are updated based on the feasibility rule. The proposed algorithm is tested on the constructed interval constrained multi-objective benchmark functions with discontinuous feasible domains and an island integrated energy system optimization scheduling problem, and is compared with three interval constrained multi-objective evolutionary algorithms. The experimental results demonstrate that the proposed algorithm has superior performance.