Research on biomedical applications of metal organic frameworks (MOFs) has recently started, and several MOFs have been identified as promising materials for drug storage. In this study, we used molecular simulations to investigate storage and release of an analgesic and anti-inflammatory drug, ibuprofen, and two cosmetic molecules, caffeine (lipo-reducer) and urea (hydrating agent) in biocompatible MOFs. We first compared the results of our molecular simulations with the experimentally available data for ibuprofen, caffeine, and urea uptakes of MOFs. Motivated from the good agreement between molecular simulations and experiments, we extended our simulations to 24 different biocompatible MOFs and predicted their ibuprofen, caffeine, and urea storage performances. Among the materials we studied, bio-MOF-100 and MOF-74 material series were identified as promising candidates for drug/cosmetic molecule storage. These biocompatible MOFs outperformed widely studied drug storage materials such as MIL-53(Fe), MIL-100(Fe), MIL-101(Cr), zeolites, and mesoporous silica (MCM-41). We also investigated diffusion of drug molecules in MOFs using molecular dynamics simulations that consider flexibility of the MOF structures. Results showed slow diffusion of drug molecules in MOFs' pores suggesting that MOFs can be strong alternatives to traditional nanoporous materials for drug storage and delivery.