Mediation analysis seeks to understand the mechanism by which a treatment affects an outcome. Count or zero‐inflated count outcomes are common in many studies in which mediation analysis is of interest. For example, in dental studies, outcomes such as the number of decayed, missing and filled teeth are typically zero inflated. Existing mediation analysis approaches for count data often assume sequential ignorability of the mediator. This is often not plausible because the mediator is not randomized so unmeasured confounders are associated with the mediator and the outcome. We develop causal methods based on instrumental variable approaches for mediation analysis for count data possibly with many 0s that do not require the assumption of sequential ignorability. We first define the direct and indirect effect ratios for those data, and then we propose estimating equations and use empirical likelihood to estimate the direct and indirect effects consistently. A sensitivity analysis is proposed for violations of the instrumental variables exclusion restriction assumption. Simulation studies demonstrate that our method works well for different types of outcome under various settings. Our method is applied to a randomized dental caries prevention trial and a study of the effect of a massive flood in Bangladesh on children's diarrhoea.