This paper considers the motion of an object subjected to a dry friction and an external random force. The objective is to characterize the role of the correlation time of the external random force. We develop efficient stochastic simulation methods for computing the diffusivity (the linear growth rate of the variance of the displacement) and other related quantities of interest when the external random force is white or colored. These methods are based on original representation formulas for the quantities of interest, which make it possible to build unbiased and consistent estimators. The numerical results obtained with these original methods are in perfect agreement with known closed-form formulas valid in the white-noise regime. In the colored-noise regime, the numerical results show that the predictions obtained from the white-noise approximation are reasonable for quantities such as the histograms of the stationary velocity but can be wrong for the diffusivity unless the correlation time is extremely small.