Machine learning is increasingly being applied to facilitate long-term, large-scale biodiversity monitoring. With most species on Earth still undiscovered or poorly documented, species-recognition models are expected to encounter new species during deployment. We introduce Open-Insects, a fine-grained image recognition benchmark dataset for open-set recognition and out-of-distribution detection in biodiversity monitoring. Open-Insects makes it possible to evaluate algorithms for new species detection on several geographical open-set splits with varying difficulty. Furthermore, we present a test set recently collected in the wild with 59 species that are likely new to science. We evaluate a variety of open-set recognition algorithms, including post-hoc methods, training-time regularization, and training with auxiliary data, finding that the simple post-hoc approach of utilizing softmax scores remains a strong baseline. We also demonstrate how to leverage auxiliary data to improve the detection performance when the training dataset is limited. Our results provide timely insights to guide the development of computer vision methods for biodiversity monitoring and species discovery.