Count data with excessive zeros are increasingly ubiquitous in genetic association studies, such as neuritic plaques in brain pathology for Alzheimer's disease. Here, we developed gene-based association tests to model such data by a mixture of two distributions, one for the structural zeros contributed by the Binomial distribution, and the other for the counts from the Poisson distribution. We derived the score statistics of the corresponding parameter of the rare variants in the zero-inflated Poisson regression model, and then constructed burden (ZIP-b) and kernel (ZIP-k) tests for the association tests. We evaluated omnibus tests that combined both ZIP-b and ZIP-k tests. Through simulated sequence data, we illustrated the potential power gain of our proposed method over a two-stage method that analyzes binary and non-zero continuous data separately for both burden and kernel tests. The ZIP burden test outperformed the kernel test as expected in all scenarios except for the scenario of variants with a mixture of directions in the genetic effects. We further demonstrated its applications to analyses of the neuritic plaque data in the ROSMAP cohort. We expect our proposed test to be useful in practice as more powerful than or complementary to the two-stage method.