Amid the growing prevalence of human-AI interaction, large language models and other AI-based entities increasingly provide forms of companionship to human users. Such AI companionship -- i.e., bonded relationships between humans and AI systems that resemble the relationships people have with family members, friends, and romantic partners -- might substantially benefit humans. Yet such relationships can also do profound harm. We propose a framework for analyzing potential negative impacts of AI companionship by identifying specific harmful traits of AI companions and speculatively mapping causal pathways back from these traits to possible causes and forward to potential harmful effects. We provide detailed, structured analysis of four potentially harmful traits -- the absence of natural endpoints for relationships, vulnerability to product sunsetting, high attachment anxiety, and propensity to engender protectiveness -- and briefly discuss fourteen others. For each trait, we propose hypotheses connecting causes -- such as misaligned optimization objectives and the digital nature of AI companions -- to fundamental harms -- including reduced autonomy, diminished quality of human relationships, and deception. Each hypothesized causal connection identifies a target for potential empirical evaluation. Our analysis examines harms at three levels: to human partners directly, to their relationships with other humans, and to society broadly. We examine how existing law struggles to address these emerging harms, discuss potential benefits of AI companions, and conclude with design recommendations for mitigating risks. This analysis offers immediate suggestions for reducing risks while laying a foundation for deeper investigation of this critical but understudied topic.