The sparse Mixture-of-Experts (MoE) has achieved significant progress for neural machine translation (NMT). However, there exist two limitations in current MoE solutions which may lead to sub-optimal performance: 1) they directly use the task knowledge of NMT into MoE (\emph{e.g.}, domain/linguistics-specific knowledge), which are generally unavailable at practical application and neglect the naturally grouped domain/linguistic properties; 2) the expert selection only depends on the localized token representation without considering the context, which fully grasps the state of each token in a global view. To address the above limitations, we propose THOR-MoE via arming the MoE with hierarchical task-guided and context-responsive routing policies. Specifically, it 1) firstly predicts the domain/language label and then extracts mixed domain/language representation to allocate task-level experts in a hierarchical manner; 2) injects the context information to enhance the token routing from the pre-selected task-level experts set, which can help each token to be accurately routed to more specialized and suitable experts. Extensive experiments on multi-domain translation and multilingual translation benchmarks with different architectures consistently demonstrate the superior performance of THOR-MoE. Additionally, the THOR-MoE operates as a plug-and-play module compatible with existing Top-$k$~\cite{shazeer2017} and Top-$p$~\cite{huang-etal-2024-harder} routing schemes, ensuring broad applicability across diverse MoE architectures. For instance, compared with vanilla Top-$p$~\cite{huang-etal-2024-harder} routing, the context-aware manner can achieve an average improvement of 0.75 BLEU with less than 22\% activated parameters on multi-domain translation tasks.