Audio-visual speech recognition (AVSR) has become critical for enhancing speech recognition in noisy environments by integrating both auditory and visual modalities. However, existing AVSR systems struggle to scale up without compromising computational efficiency. In this study, we introduce MoHAVE (Mixture of Hierarchical Audio-Visual Experts), a novel robust AVSR framework designed to address these scalability constraints. By leveraging a Mixture-of-Experts (MoE) architecture, MoHAVE activates modality-specific expert groups, ensuring dynamic adaptation to various audio-visual inputs with minimal computational overhead. Key contributions of MoHAVE include: (1) a sparse MoE framework that efficiently scales AVSR model capacity, (2) a hierarchical gating mechanism that dynamically utilizes the expert groups based on input context, enhancing adaptability and robustness, and (3) remarkable performance across robust AVSR benchmarks, including LRS3 and MuAViC transcription and translation tasks, setting a new standard for scalable speech recognition systems.