Reliable data collection, transmission, and delivery on Internet of Things (IoT) systems is crucial in order to provide high-quality intelligent services. However, sensor data delivery can be interrupted for various reasons, such as sensor malfunction, network failures, and external attacks. Thus, only data from a partial set of sensors may be available. We call it the missing sensor problem. This problem can lead to severe performance degradation at inference time by neural-network-based recognition models trained on the complete sensor set. This paper enhances the robustness of neural network models to the missing sensor problem by introducing a novel feature reconstruction module, named the graph recovery module, that handles missing sensors directly inside the network. Specifically, we consider topology-aware IoT applications, where sensors are placed on a physically interconnected network. We design a novel neural message passing mechanism that logically mimics physical network topology, based on recent advances in graph neural networks (GNNs). We rely on a spatial locality assumption, where only correlations between physically connected sensors are explicitly explored. When encountering missing sensors, information is passed from available sensors to missing sensors to be used to reconstruct their features. Moreover, at each message passing step, we utilize a gating mechanism inspired by Gated Recurrent Units (GRUs) to automatically control information flow between available sensors and missing sensors. We empirically evaluate the reconstruction performance of the graph recovery module with two representative IoT applications; human activity recognition (HAR) and electroencephalogram (EEG)-based motor-imagery classification, on three public datasets. Two different backbone networks are utilized for the tasks. Our design is shown to effectively maintain model performance, suffering only 7% to 18% accuracy loss when as much as 90% of sensors are removed, compared to a drop of 15% to 47% in the accuracy of competing state-of-the-art algorithms under the same conditions. The accuracy gap is largest when more sensors are missing.