We propose a geometric framework to assess global sensitivity of Bayesian nonparametric models in density estimation. Our measures build upon the nonparametric Fisher-Rao Riemannian metric which, under the square-root transform of probability density functions, provides computationally efficient tools for exploring variability in posterior samples of densities, as well as calculating their averages, geodesic paths and distances. We consider models for density estimation based on Dirichlet-type priors and perform sensitivity analysis by perturbing either the precision parameter or the base probability measure. To determine the different effects of the perturbations of the parameters and hyperparameters in the models on the posterior, we define four geometric complementary global sensitivity measures: (1) the Fisher-Rao distance between density averages of posterior samples, (2) difference in overall Karcher variances of posterior samples, (3) L2 norm of difference in scaled eigenvalues of covariance matrices obtained from posterior samples and (4) Wasserstein-type distance between posterior samples. We validate our approach using multiple simulation studies and real datasets.