Accurate temperature measurements are essential for the proper monitoring and control of industrial furnaces. However, measurement uncertainty is a risk for such a critical parameter. Certain instrumental and environmental errors must be considered when using spectral-band radiation thermometry techniques, such as the uncertainty in the emissivity of the target surface, reflected radiation from surrounding objects, or atmospheric absorption and emission, to name a few. Undesired contributions to measured radiation can be isolated using measurement models, also known as error-correction models. This paper presents a methodology for budgeting significant sources of error and uncertainty during temperature measurements in a petrochemical furnace scenario. A continuous monitoring system is also presented, aided by a deep-learning-based measurement correction model, to allow domain experts to analyze the furnace's operation in real-time. To validate the proposed system's functionality, a real-world application case in a petrochemical plant is presented. The proposed solution demonstrates the viability of precise industrial furnace monitoring, thereby increasing operational security and improving the efficiency of such energy-intensive systems.