Recent advances in diffusion models have significantly enhanced image generation capabilities. However, customizing these models with new classes often leads to unintended consequences that compromise their reliability. We introduce the concept of open-world forgetting to characterize the vast scope of these unintended alterations. Our work presents the first systematic investigation into open-world forgetting in diffusion models, focusing on semantic and appearance drift of representations. Using zero-shot classification, we demonstrate that even minor model adaptations can lead to significant semantic drift affecting areas far beyond newly introduced concepts, with accuracy drops of up to 60% on previously learned concepts. Our analysis of appearance drift reveals substantial changes in texture and color distributions of generated content. To address these issues, we propose a functional regularization strategy that effectively preserves original capabilities while accommodating new concepts. Through extensive experiments across multiple datasets and evaluation metrics, we demonstrate that our approach significantly reduces both semantic and appearance drift. Our study highlights the importance of considering open-world forgetting in future research on model customization and finetuning methods.