In this paper, we consider inference and uncertainty quantification for low Tucker rank tensors with additive noise in the high-dimensional regime. Focusing on the output of the higher-order orthogonal iteration (HOOI) algorithm, a commonly used algorithm for tensor singular value decomposition, we establish non-asymptotic distributional theory and study how to construct confidence regions and intervals for both the estimated singular vectors and the tensor entries in the presence of heteroskedastic subgaussian noise, which are further shown to be optimal for homoskedastic Gaussian noise. Furthermore, as a byproduct of our theoretical results, we establish the entrywise convergence of HOOI when initialized via diagonal deletion. To further illustrate the utility of our theoretical results, we then consider several concrete statistical inference tasks. First, in the tensor mixed-membership blockmodel, we consider a two-sample test for equality of membership profiles, and we propose a test statistic with consistency under local alternatives that exhibits a power improvement relative to the corresponding matrix test considered in several previous works. Next, we consider simultaneous inference for small collections of entries of the tensor, and we obtain consistent confidence regions. Finally, focusing on the particular case of testing whether entries of the tensor are equal, we propose a consistent test statistic that shows how index overlap results in different asymptotic standard deviations. All of our proposed procedures are fully data-driven, adaptive to noise distribution and signal strength, and do not rely on sample-splitting, and our main results highlight the effect of higher-order structures on estimation relative to the matrix setting. Our theoretical results are demonstrated through numerical simulations.