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Model Predictive Control (MPC) has demonstrated significant potential in improving energy efficiency in building climate control, outperforming traditional controllers commonly used in modern building management systems. Among MPC variants, Data-driven Predictive Control (DPC) offers the advantage of modeling building dynamics directly from data, thereby substantially reducing commissioning efforts. However, inevitable model uncertainties and measurement noise can result in comfort violations, even with dedicated MPC setups. This paper introduces a Disturbance-Adaptive DPC (DAD-DPC) framework that ensures asymptotic satisfaction of predefined violation bounds without knowing the uncertainty and noise distributions. The framework employs a data-driven pipeline based on Willems' Fundamental Lemma and conformal prediction for application in building climate control. The proposed DAD-DPC framework was validated through four building cases using the high-fidelity BOPTEST simulation platform and an occupied campus building, Polydome. DAD-DPC successfully regulated the average comfort violations to meet pre-defined bounds. Notably, the 5%-violation DAD-DPC setup achieved 30.1%/11.2%/27.1%/20.5% energy savings compared to default controllers across four cases. These results demonstrate the framework's effectiveness in balancing energy consumption and comfort violations, offering a practical solution for building climate control applications.