Recent breakthroughs in large language models (LLMs) have generated both interest and concern about their potential adoption as information sources or communication tools across different domains. In public health, where stakes are high and impacts extend across diverse populations, adopting LLMs poses unique challenges that require thorough evaluation. However, structured approaches for assessing potential risks in public health remain under-explored. To address this gap, we conducted focus groups with public health professionals and individuals with lived experience to unpack their concerns, situated across three distinct and critical public health issues that demand high-quality information: infectious disease prevention (vaccines), chronic and well-being care (opioid use disorder), and community health and safety (intimate partner violence). We synthesize participants' perspectives into a risk taxonomy, identifying and contextualizing the potential harms LLMs may introduce when positioned alongside traditional health communication. This taxonomy highlights four dimensions of risk to individuals, human-centered care, information ecosystem, and technology accountability. For each dimension, we unpack specific risks and offer example reflection questions to help practitioners adopt a risk-reflexive approach. By summarizing distinctive LLM characteristics and linking them to identified risks, we discuss the need to revisit prior mental models of information behaviors and complement evaluations with external validity and domain expertise through lived experience and real-world practices. Together, this work contributes a shared vocabulary and reflection tool for people in both computing and public health to collaboratively anticipate, evaluate, and mitigate risks in deciding when to employ LLM capabilities (or not) and how to mitigate harm.