Large language models (LLMs) are already being piloted for clinical use in hospital systems like NYU Langone, Dana-Farber and the NHS. A proposed deployment use case is psychotherapy, where a LLM-powered chatbot can treat a patient undergoing a mental health crisis. Deployment of LLMs for mental health response could hypothetically broaden access to psychotherapy and provide new possibilities for personalizing care. However, recent high-profile failures, like damaging dieting advice offered by the Tessa chatbot to patients with eating disorders, have led to doubt about their reliability in high-stakes and safety-critical settings.In this work, we develop an evaluation framework for determining whether LLM response is a viable and ethical path forward for the automation of mental health treatment. Our framework measures equity in empathy and adherence of LLM responses to motivational interviewing theory. Using human evaluation with trained clinicians and automatic quality-of-care metrics grounded in psychology research, we compare the responses provided by peer-to-peer responders to those provided by a state-of-the-art LLM.We show that LLMs like GPT-4 use implicit and explicit cues to infer patient demographics like race. We then show that there are statistically significant discrepancies between patient subgroups: Responses to Black posters consistently have lower empathy than for any other demographic group (2%-13% lower than the control group). Promisingly, we do find that the manner in which responses are generated significantly impacts the quality of the response. We conclude by proposing safety guidelines for the potential deployment of LLMs for mental health response.