An important task in the Internet of Things (IoT) is field monitoring, where multiple IoT nodes take measurements and communicate them to the base station or the cloud for processing, inference, and analysis. When the measurements are high-dimensional (e.g., videos or time-series data), IoT networks with limited bandwidth and low-power devices may not be able to support such frequent transmissions with high data rates. To ensure communication efficiency, this article proposes to model the measurement compression at IoT nodes and the inference at the base station or cloud as a deep neural network (DNN). We propose a new framework where the data to be transmitted from nodes are the intermediate outputs of a layer of the DNN. We show how to learn the model parameters of the DNN and study the trade-off between the communication rate and the inference accuracy. The experimental results show that we can save approximately 96 percent transmissions with only a degradation of 2.5 percent in inference accuracy, which shows the potentiality to enable many new IoT data analysis applications that generate a large amount of measurements.