Temporal interaction graphs (TIGs), defined by sequences of timestamped interaction events, have become ubiquitous in real-world applications due to their capability to model complex dynamic system behaviors. As a result, temporal interaction graph representation learning (TIGRL) has garnered significant attention in recent years. TIGRL aims to embed nodes in TIGs into low-dimensional representations that effectively preserve both structural and temporal information, thereby enhancing the performance of downstream tasks such as classification, prediction, and clustering within constantly evolving data environments. In this paper, we begin by introducing the foundational concepts of TIGs and emphasize the critical role of temporal dependencies. We then propose a comprehensive taxonomy of state-of-the-art TIGRL methods, systematically categorizing them based on the types of information utilized during the learning process to address the unique challenges inherent to TIGs. To facilitate further research and practical applications, we curate the source of datasets and benchmarks, providing valuable resources for empirical investigations. Finally, we examine key open challenges and explore promising research directions in TIGRL, laying the groundwork for future advancements that have the potential to shape the evolution of this field.