Edge AI is the fusion of edge computing and artificial intelligence (AI). It promises responsiveness, privacy preservation, and fault tolerance by moving parts of the AI workflow from centralized cloud data centers to geographically dispersed edge servers, which are located at the source of the data. The scale of edge AI can vary from simple data preprocessing tasks to the whole machine learning stack. However, most edge AI implementations so far are limited to urban areas, where the infrastructure is highly dependable. This work instead focuses on a class of applications involved in environmental monitoring in remote, rural areas such as forests and rivers. Such applications have additional challenges, including failure proneness and access to the electricity grid and communication networks. We propose neuromorphic computing as a promising solution to the energy, communication, and computation constraints in such scenarios and identify directions for future research in neuromorphic edge AI for rural environmental monitoring. Proposed directions are distributed model synchronization, edge-only learning, aerial networks, spiking neural networks, and sensor integration.