The development of new reactor technologies requires careful assessments of the various sources of epistemic uncertainties. In this work, nuclear data uncertainties featuring the main isotopes of the U/Th molten salt fast reactor (MSFR) design are propagated through Monte Carlo calculations to quantify the final uncertainty on some relevant integral parameters. In the first part of this paper, some best-estimate calculations are performed by selecting different nuclear data libraries, showing the remarkable impact of this choice on the final responses. Then the Generalized Perturbation Theory routine available in Serpent 2 is adopted for a preliminary sensitivity and uncertainty analyses with respect to keff, highlighting a significant discrepancy between the covariance of the JEFF-3.3 and ENDF/B-VIII.0 libraries. After the selection of a few relevant nuclides, namely, 7Li, 19F, 232Th, and 233U, the Total Monte Carlo method and the unscented transform (UT) are then adopted to estimate the uncertainty of other responses of interest like the conversion ratio and some multigroup constants. Some potential issues of the UT are highlighted, and a mitigation strategy is applied. A relevant result of this analysis concerns the need for better data evaluations for the nuclides constituting the circulating salt for an effective deployment of the MSFR technology.