Existing brain tumor segmentation methods usually utilize multiple Magnetic Resonance Imaging (MRI) modalities in brain tumor images for segmentation, which can achieve better segmentation performance. However, in clinical applications, some modalities are often missing due to resource constraints, resulting in significant performance degradation for methods that rely on complete modality segmentation. In this paper, we propose a Multimodal feature distillation with Mamba-Transformer hybrid network (MMTSeg) for accurate brain tumor segmentation with missing modalities. We first employ a Multimodal Feature Distillation (MFD) module to distill feature-level multimodal knowledge into different unimodalities to extract complete modality information. We further develop an Unimodal Feature Enhancement (UFE) module to model the semantic relationship between global and local information. Finally, we built a Cross-Modal Fusion (CMF) module to explicitly align the global correlations across modalities, even when some modalities are missing. Complementary features within and across modalities are refined by the Mamba-Transformer hybrid architectures in both the UFE and CMF modules, dynamically capturing long-range dependencies and global semantic information for complex spatial contexts. A boundary-wise loss function is employed as the segmentation loss of the proposed MMTSeg to minimize boundary discrepancies for a distance-based metric. Our ablation study demonstrates the importance of the proposed feature enhancement and fusion modules in the proposed network and the Transformer with Mamba block for improving the performance of brain tumor segmentation with missing modalities. Extensive experiments on the BraTS 2018 and BraTS 2020 datasets demonstrate that the proposed MMTSeg framework outperforms state-of-the-art methods when modalities are missing.