The cross-match test is an exact, distribution-free test of no treatment effect on a high-dimensional outcome in a randomized experiment. The test uses optimal nonbipartite matching to pair 2I subjects into I pairs based on similar outcomes, and the cross-match statistic A is the number of times that a treated subject was paired with a control, rejecting for small values of A. If the test is applied in an observational study in which treatments are not randomly assigned, then it may be comparing treated and control subjects who are not comparable, and thus may falsely reject a true null hypothesis of no treatment effect. We develop a sensitivity analysis for the cross-match test and apply it in an observational study of the effects of smoking on gene expression levels. In addition, we develop a sensitivity analysis for several multiple testing procedures using the cross-match test and apply it to 1627 molecular function categories in Gene Ontology.