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Recent advances in Internet of Things (IoT) technologies have sparked significant interest toward developing learning-based sensing applications on embedded edge devices. These efforts, however, are being challenged by the complexities of adapting to unforeseen conditions in an open-world environment, mainly due to the intensive computational and energy demands exceeding the capabilities of edge devices. In this article, we propose OpenSense, an open-world time-series sensing framework for making inferences from time-series sensor data and achieving incremental learning on an embedded edge device with limited resources. The proposed framework is able to achieve two essential tasks, inference and incremental learning, eliminating the necessity for powerful cloud servers. In addition, to secure enough time for incremental learning and reduce energy consumption, we need to schedule sensing activities without missing any events in the environment. Therefore, we propose two dynamic sensor scheduling techniques: 1) a class-level period assignment scheduler that finds an appropriate sensing period for each inferred class and 2) a Q-learning-based scheduler that dynamically determines the sensing interval for each classification moment by learning the patterns of event classes. With this framework, we discuss the design choices made to ensure satisfactory learning performance and efficient resource usage. Experimental results demonstrate the ability of the system to incrementally adapt to unforeseen conditions and to efficiently schedule to run on a resource-constrained device.