The ubiquitous data streaming through the Internet of Things (IoT) creates security risks. Intrusion detection systems (IDS) based on machine learning can support user security. Generative Artificial Intelligence (GenAI) demonstrates strong capabilities in generating synthetic data based on realistic distributions and learning complex patterns from high-dimensional data. By harnessing the capabilities of generative AI, it is feasible to augment intrusion detection models, allowing for more robust and adaptive security solutions in IoT environments. This paper introduces a systematic literature review of recent GenAI applications in IoT IDS and analyzes the architectures and techniques in the models. We classify the common usages such as data augmentation and class balancing, data reconstruction, and adversarial attack generation. We outline the commonly used datasets and evaluation metrics and compare the performances of each model under these conditions. The study identifies current challenges and emerging research trends in various technologies for applying GenAI in IoT IDS.