The impacts of climate change are anticipated to become stronger in the future, leading to higher costs and more severe accidents in the oil industry’s facilities and surrounding communities. Motivated by this, the main objective of this paper is to develop, for the oil industry, a risk assessment methodology that considers future climate projections. In the context of an action research effort, carried out in a refinery in Greece, we adapted the organization’s extant risk management approach based on the Risk Assessment Matrix (RAM) and suggested a risk quantification process that incorporates future climate projections. The Climate Risk Assessment Matrix (CRAM) was developed to be used to assess the exposure of the facility’s assets, including human resources, to future climate risks. To evaluate CRAM, a comparison with RAM for the specific organization for the period 1980–2004 was made. Next, the application of CRAM for the period 2025–2049 indicated that, even though the resilience of the operations of the company to extreme conditions seems adequate at present, increased attention should be paid in the future to the resilience of refinery processes, the cooling system, and human resources. Beyond the specific case, the paper provides lessons for similar organizations and infrastructures located elsewhere.