Unfairness is a well-known challenge in Recommender Systems (RSs), often resulting in biased outcomes that disadvantage users or items based on attributes such as gender, race, age, or popularity. Although some approaches have started to improve fairness recommendation in offline or static contexts, the issue of unfairness often exacerbates over time, leading to significant problems like the Matthew effect, filter bubbles, and echo chambers. To address these challenges, we proposed a novel framework, Hypergraph Contrastive Multi-Interest Learning for Fair Conversational Recommender System (HyFairCRS), aiming to promote multi-interest diversity fairness in dynamic and interactive Conversational Recommender Systems (CRSs). HyFairCRS first captures a wide range of user interests by establishing diverse hypergraphs through contrastive learning. These interests are then utilized in conversations to generate informative responses and ensure fair item predictions within the dynamic user-system feedback loop. Experiments on two CRS-based datasets show that HyFairCRS achieves a new state-of-the-art performance while effectively alleviating unfairness. Our code is available at https://github.com/zysensmile/HyFairCRS.