Dialogue systems in the form of chatbots and personal assistants are being increasingly integrated into people's lives. These dialogue systems often have the ability to adopt an anthropomorphic persona, mimicking a societal demographic to appear more approachable and trustworthy to users. However, the adoption of a persona can result in the adoption of biases. We define persona biases as harmful differences in text (e.g., varying levels of offensiveness or affirmations of biased statements) generated from adopting different demographic personas. In this paper, we present the first large-scale study on persona biases in dialogue systems and conduct analyses on personas of different social classes, sexual orientations, races, and genders. Furthermore, we introduce an open-source framework, UnitPersonaBias, a tool to explore and aggregate subtle persona biases in dialogue systems. In our studies of the Blender and DialoGPT dialogue systems, we show that the choice of personas can affect the degree of harms in generated responses. Additionally, adopting personas of more diverse, historically marginalized demographics appears to decrease harmful responses the most.