A large body of research has found substantial gender bias in NLP systems. Most of this research takes a binary, essentialist view of gender: limiting its variation to the categories _men_ and _women_, conflating gender with sex, and ignoring different sexual identities. But gender and sexuality exist on a spectrum, so in this paper we study the biases of large language models (LLMs) towards sexual and gender minorities beyond binary categories. Grounding our study in a widely used social psychology model -- the Stereotype Content Model -- we demonstrate that English-language survey questions about social perceptions elicit more negative stereotypes of sexual and gender minorities from both humans and LLMs. We then extend this framework to a more realistic use case: text generation. Our analysis shows that LLMs generate stereotyped representations of sexual and gender minorities in this setting, showing that they amplify representational harms in creative writing, a widely advertised use for LLMs.