Preserving privacy in blockchain-based systems is crucial for ensuring anonymity and confidentiality during transactions. While cryptographic solutions can address on-chain privacy concerns, their implementation on blockchains may introduce performance overhead, which remains unclear to researchers and practitioners. This paper investigates the performance impact of integrating zero-knowledge proofs (ZKPs) into the widely adopted permissioned blockchain framework called Hyperledger Fabric. The study focuses on evaluating the scalability and bottleneck aspects of blockchain platforms incorporating ZKPs. Through comprehensive experimentation and analysis, the study reveals that the integration of ZKPs compromises performance in terms of transaction rates and latency, while effectively safeguarding users' personal information. Implementing on-chain ZKP feature would result in a performance loss of 30% to 87.5% in various experimental configurations in Hyperledger Fabric. The findings presented in this paper are informative for the design and implementation of blockchain-based systems with strict privacy requirements.