Machine Unlearning (MU) aims to remove the influence of specific knowledge from a pretrained model. Existing methods often rely on retained training data to preserve utility; such dependence is impractical due to privacy and scalability constraints. A further complication arises when unlearning is applied to vision-language models (VLMs), where entangled multimodal representations make targeted forgetting especially challenging. We propose DIET, a principled retain-data-free unlearning method for VLMs that addresses these challenges by leveraging the geometry of hyperbolic space. The core idea is to push forget embeddings toward class-mismatched prototypes located at the boundary of the hyperbolic space. In hyperbolic geometry, points near the boundary become infinitely distant from interior points. As a result, moving forget embeddings to the boundary makes their influence on the model asymptotically negligible. To formalize this, we guide the forgetting process using the Busemann function, which quantifies directional distance to the boundary. We further develop an adaptive scheme based on optimal transport that selects mismatched prototypes for each forget embedding, enabling flexible unlearning dynamics. Extensive experiments on fine-grained datasets such as Flowers102, OxfordPets, and StanfordCars show that DIET achieves an average forget accuracy of 8.06\%, while preserving 69.04\% utility using only 16 samples per concept, significantly outperforming the best retain-free baselines with a 117.5\% in model utility, and showing competitive performance to retain-data baselines with only a 3.79\% drop