In observational studies, contingency tables are commonly used to examine associations between categorical variables. However, any test of association in contingency tables may be biased by unmeasured confounding, and existing sensitivity analyses typically assume a binary treatment or impose strong parametric assumptions on non-binary treatments. We develop an exact (non-asymptotic) and nonparametric sensitivity analysis for unmeasured confounding in $I \times J$ and $I\times J\times K$ contingency tables, accommodating both non-binary treatments and outcomes. Extending Rosenbaum's generic bias sensitivity model, we derive a general method to compute the exact worst-case null distribution for any count-based test, including chi-squared and likelihood-based tests of association. We further provide specialized results for subfamilies of count-based tests that enable more efficient computation of the worst-case null distribution. Finally, we investigate test power in sensitivity analyses and show that tests exploiting all treatment and outcome levels achieve higher power than tests that dichotomize the categorical variables. We illustrate the proposed methods with a re-analysis of the effect of pre-kindergarten care on math achievement using data from the Early Childhood Longitudinal Study. The R package sensitivityIxJ that implements the method is on CRAN.

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