Audio-based disease prediction is emerging as a promising supplement to traditional medical diagnosis methods, facilitating early, convenient, and non-invasive disease detection and prevention. Multimodal fusion, which integrates features from various domains within or across bio-acoustic modalities, has proven effective in enhancing diagnostic performance. However, most existing methods in the field employ unilateral fusion strategies that focus solely on either intra-modal or inter-modal fusion. This approach limits the full exploitation of the complementary nature of diverse acoustic feature domains and bio-acoustic modalities. Additionally, the inadequate and isolated exploration of latent dependencies within modality-specific and modality-shared spaces curtails their capacity to manage the inherent heterogeneity in multimodal data. To fill these gaps, we propose a transformer-based hierarchical fusion network designed for general multimodal audio-based disease prediction. Specifically, we seamlessly integrate intra-modal and inter-modal fusion in a hierarchical manner and proficiently encode the necessary intra-modal and inter-modal complementary correlations, respectively. Comprehensive experiments demonstrate that our model achieves state-of-the-art performance in predicting three diseases: COVID-19, Parkinson's disease, and pathological dysarthria, showcasing its promising potential in a broad context of audio-based disease prediction tasks. Additionally, extensive ablation studies and qualitative analyses highlight the significant benefits of each main component within our model.