We study a hypothesis testing problem with a privacy constraint over a noisy channel and derive the performance of optimal tests under the Neyman-Pearson criterion. The fundamental limit of interest is the privacy-utility tradeoff (PUT) between the exponent of the type-II error probability and the leakage of the information source subject to a constant constraint on the type-I error probability. We provide an exact characterization of the asymptotic PUT for any non-vanishing type-I error probability. Our result implies that tolerating a larger type-I error probability cannot improve the PUT. Such a result is known as a strong converse or strong impossibility theorem. To prove the strong converse theorem, we apply the recently proposed technique in (Tyagi and Watanabe, 2020) and further demonstrate its generality. The strong converse theorems for several problems, such as hypothesis testing against independence over a noisy channel (Sreekumar and Gündüz, 2020) and hypothesis testing with communication and privacy constraints (Gilani \emph{et al.}, 2020), are established or recovered as special cases of our result.