Assessing causal treatment effect on a time-to-event outcome is of key interest in many scientific investigations. Instrumental variable (IV) is a useful tool to mitigate the impact of endogenous treatment selection to attain unbiased estimation of causal treatment effect. Existing development of IV methodology, however, has not attended to outcomes subject to interval censoring, which are ubiquitously present in studies with intermittent follow-up but are challenging to handle in terms of both theory and computation. In this work, we fill in this important gap by studying a general class of causal semiparametric transformation models with interval-censored data. We propose a nonparametric maximum likelihood estimator of the complier causal treatment effect. Moreover, we design a reliable and computationally stable expectation-maximization (EM) algorithm, which has a tractable objective function in the maximization step via the use of Poisson latent variables. The asymptotic properties of the proposed estimators, including the consistency, asymptotic normality, and semiparametric efficiency, are established with empirical process techniques. We conduct extensive simulation studies and an application to a colorectal cancer screening data set, showing satisfactory finite-sample performance of the proposed method as well as its prominent advantages over naive methods.