The field of Natural Language Generation (NLG) suffers from a severe shortage of labeled data due to the extremely expensive and time-consuming process involved in manual annotation. A natural approach for coping with this problem is active learning (AL), a well-known machine learning technique for improving annotation efficiency by selectively choosing the most informative examples to label. However, while AL has been well-researched in the context of text classification, its application to NLG remains largely unexplored. In this paper, we present a first systematic study of active learning for NLG, considering a diverse set of tasks and multiple leading selection strategies, and harnessing a strong instruction-tuned model. Our results indicate that the performance of existing AL strategies is inconsistent, surpassing the baseline of random example selection in some cases but not in others. We highlight some notable differences between the classification and generation scenarios, and analyze the selection behaviors of existing AL strategies. Our findings motivate exploring novel approaches for applying AL to generation tasks.