The problem of calculating sensitivity coefficients to nuclear data in burnup problems is addressed. A non-intrusive method based on the random sampling of nuclear data and on linear regression is proposed to overcome the limitations of dealing with a complex coupled problem. Sensitivity plots are reported for the actinide concentrations that build up in irradiated fuel and that have the largest impact for the safe and economical management of spent nuclear fuel. Despite using a simplified pincell model, the range of applicability of the calculated sensitivities is further discussed. The sensitivity study is corroborated by an analysis of how different nuclear data libraries perform in the prediction of the neutron multiplication factor in a burnup problem. The origin of the discrepancies between the results obtained using the JEFF-3.3 and ENDF/B-VIII.0 nuclear data libraries are investigated.