Assessing sensitivity to unmeasured confounding is an important step in observational studies, which typically estimate effects under the assumption that all confounders are measured. In this paper, we develop a sensitivity analysis framework for balancing weights estimators, an increasingly popular approach that solves an optimization problem to obtain weights that directly minimizes covariate imbalance. In particular, we adapt a sensitivity analysis framework using the percentile bootstrap for a broad class of balancing weights estimators. We prove that the percentile bootstrap procedure can, with only minor modifications, yield valid confidence intervals for causal effects under restrictions on the level of unmeasured confounding. We also propose an amplification--a mapping from a one-dimensional sensitivity analysis to a higher dimensional sensitivity analysis--to allow for interpretable sensitivity parameters in the balancing weights framework. We illustrate our method through extensive real data examples. [This paper will be published in "Journal of the Royal Statistical Society Series A: Statistics in Society."]