To provide reliable safety margins in reactor design and safety analysis, the best estimate plus uncertainty (BEPU) analysis, which is recommended by the International Atomic Energy Agency (IAEA), has drawn increasing attention worldwide. In order to systematically evaluate the sensitivity and uncertainty in the design and safety analysis of molten salt reactors (MSRs), a sensitivity and uncertainty analysis framework has been developed by integrating the reactor system safety analysis code RELAP5-TMSR with the data analysis code RAVEN. The framework is tested using the transient scenarios of the molten salt reactor experiment (MSRE): reactivity insertion accident (RIA) and station blackout (SBO). The testing results demonstrate that the proposed framework effectively conducts sensitivity and uncertainty analysis. Sensitivity analyses identify key input parameters, including the primary exchanger parameters, air radiator parameters, initial temperatures, delayed neutron parameters and volumetric heat capacity of the INOR-8 alloy. Uncertainty quantification provides 95% confidence intervals for the figures of merit (FOMs) and the steady-state and RIA scenarios remained within safety limits. The developed framework enables automated, efficient, and high-capacity sensitivity and uncertainty analysis across multiple parameters and transient scenarios. The systematic analysis provides sensitivity indicators and uncertainty distributions, offering quantitative insights into the safety margins and supporting the design and safety analysis of MSRs.