Laser pipe-cutting system, a special machine tool, has been widely used in the precision machining of metal pipe. The geometric errors remarkably affect the machining accuracy of products. Error modeling and sensitivity analysis are key issues to improve the product quality. In this paper, the geometric error model of the laser pipe-cutting system which contains 70 geometric errors is established based on multi-body theory and Lie group. The coupling effects caused by two chucks are considered as the spatial angular deviation in modeling. The sensitivity analysis for the geometric error model is conducted to identify the essential sensitivity errors based on an improved Sobol method with quasi-Monte Carlo algorithm. The results show that not only the linear positioning errors, but also the squareness errors and parallelism errors play crucial roles in the machining accuracy. Based on the result, the essential sensitivity errors are calibrated and the machining accuracy is improved.