There are many real-world applications with uncertainties that can be modeled as dynamic interval multi-objective optimization problems (DI-MOPs). However, it is challenging for traditional algorithms to converge rapidly before time-varying parameters change and to obtain optimal solutions under interval objectives. So far, there is a lack of studies on evaluation methods for interval optimal solutions in dynamic problems. Therefore, a fast evaluation framework is proposed in this paper to tackle these issues. In this framework, we first derive a new hash function based on the Canberra distance and provide a theoretical proof of the validity and local sensitivity of the hash function, from which a Canberra Locality Sensitive Hashing (CLSH) is constructed. The CLSH accelerates the search for interval evaluation objects in uncertain environments. Further, we propose an Adaptive Interval Crowding Distance (AICD) with relaxed constraints to obtain a global improvement in the quality of the solutions. The candidate solutions in the above framework are generated by the environment awareness and directed migration of the muti-objective bacteria colony chemotaxis (MOBCC) algorithm. This complete algorithm is called the dynamic interval MOBCC (DI-MOBCC). In addition, theoretical proofs of the validity and local sensitivity of hash functions are also provided. Computational results on eight benchmark optimization problems and a path planning of mobile robots in uncertain environments validate that DI-MOBCC is more competitive than other state-of-the-art algorithms in tackling DI-MOPs.