Considering physical sensors with certain sensing capabilities in an Internet-of-Things (IoTs) sensory environment, in this paper, we propose an efficient energy management framework to control the duty cycles of these sensors under quality-of-information (QoI) expectations in a multitask-oriented environment. Contrary to past research efforts, our proposal is transparent and compatible both with the underlying low-layer protocols and diverse applications, and preserving energy-efficiency in the long run without sacrificing the QoI levels attained. In particular, we first introduce the novel concept of QoI-aware sensor-to-task relevancy to explicitly consider the sensing capabilities offered by a sensor to the IoT sensory environments, and QoI requirements required by a task. Second, we propose a novel concept of the critical covering set of any given task in selecting the sensors to service a task over time. Third, energy management decision is made dynamically at runtime, to reach the optimum for long-term application arrivals and departures under the constraint of their service delay. We show a case study to utilize sensors to perform environmental monitoring with a complete set of performance analysis. We further consider the signal propagation and processing latency into the proposal, and provide a thorough analysis on its impact on average measured delay probability.