We study mean change point testing problems for high-dimensional data, with exponentially- or polynomially-decaying tails. In each case, depending on the $\ell_0$-norm of the mean change vector, we separately consider dense and sparse regimes. We characterise the boundary between the dense and sparse regimes under the above two tail conditions for the first time in the change point literature and propose novel testing procedures that attain optimal rates in each of the four regimes up to a poly-iterated logarithmic factor. By comparing with previous results under Gaussian assumptions, our results quantify the costs of heavy-tailedness on the fundamental difficulty of change point testing problems for high-dimensional data. To be specific, when the error distributions possess exponentially-decaying tails, a CUSUM-type statistic is shown to achieve a minimax testing rate up to $\sqrt{\log\log(8n)}$. As for polynomially-decaying tails, admitting bounded $α$-th moments for some $α\geq 4$, we introduce a median-of-means-type test statistic that achieves a near-optimal testing rate in both dense and sparse regimes. In the sparse regime, we further propose a computationally-efficient test to achieve optimality. Our investigation in the even more challenging case of $2 \leq α< 4$, unveils a new phenomenon that the minimax testing rate has no sparse regime, i.e.\ testing sparse changes is information-theoretically as hard as testing dense changes. Finally, we consider various extensions where we also obtain near-optimal performances, including testing against multiple change points, allowing temporal dependence as well as fewer than two finite moments in the data generating mechanisms. We also show how sub-Gaussian rates can be achieved when an additional minimal spacing condition is imposed under the alternative hypothesis.