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In this paper, we propose a task scheduling strategy for an Internet of Things (IoT) node powered by renewable energy (RE). The node is assumed to have a rechargeable battery and an RE harvester. Moreover, the node is requested to perform M tasks over a planning period of N >= M time slots. For each task, a priority rating and a reward are assigned. With these considerations we develop a mathematical framework to optimize the utility of the node, defined as the sum of rewards over the specified planning horizon. Using the proposed framework, we derive a genie-aided strategy, which serves as a performance benchmark for online algorithms. We then propose two online task scheduling strategies of different complexity level, which correspond to a Mixed Integer Linear Programming (MILP) based approach and later on, a simpler sorting-based mechanism is also introduced. The presented techniques use existing forecasting methods to estimate future RE production. We finally evaluate the performance of the proposed strategies and their robustness to forecasting errors through extensive simulations. The impact of system parameters such as battery size and RE harvesting capacity are also examined numerically.