Advances in large language models (LLMs) are profoundly reshaping the field of human-robot interaction (HRI). While prior work has highlighted the technical potential of LLMs, few studies have systematically examined their human-centered impact (e.g., human-oriented understanding, user modeling, and levels of autonomy), making it difficult to consolidate emerging challenges in LLM-driven HRI systems. Therefore, we conducted a systematic literature search following the PRISMA guideline, identifying 86 articles that met our inclusion criteria. Our findings reveal that: (1) LLMs are transforming the fundamentals of HRI by reshaping how robots sense context, generate socially grounded interactions, and maintain continuous alignment with human needs in embodied settings; and (2) current research is largely exploratory, with different studies focusing on different facets of LLM-driven HRI, resulting in wide-ranging choices of experimental setups, study methods, and evaluation metrics. Finally, we identify key design considerations and challenges, offering a coherent overview and guidelines for future research at the intersection of LLMs and HRI.