The integration of Generative Artificial Intelligence (GenAI) within the Internet of Things (IoT) is garnering considerable interest. This growing attention stems from the continuous evolution and widespread adoption they are both having individually, enough to spontaneously reshape numerous sectors, including Healthcare, Manufacturing, and Smart Cities. Hence, their increasing popularity has catalyzed further extensive research for understanding the potential of the duo GenAI-IoT, how they interplay, and to which extent their synergy can innovate the state-of-the-art in their individual scenarios. However, despite the increasing prominence of GenAI for IoT Computing, much of the existing research remains focused on specific, narrowly scoped applications. This fragmented approach highlights the need for a more comprehensive analysis of the potential, challenges, and implications of GenAI integration within the broader IoT ecosystem. This survey exactly aims to address this gap by providing a holistic overview of the opportunities, issues, and considerations arising from the convergence of these mainstream paradigms. Our contribution is realized through a systematic literature review following the PRISMA methodology. A comparison framework is presented, and well-defined research questions are outlined to comprehensively explore the past, present, and future directions of GenAI integration with IoT Computing, offering valuable insights for both experts and newcomers.

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