We obtain estimates for the Kolmogorov distance to appropriately chosen gaussians, of linear functions \[ \sum_{i\in [n]^d} θ_i X_i \] of random tensors $\boldsymbol{X}=\langle X_i:i\in [n]^d\rangle$ which are symmetric and exchangeable, and whose entries have bounded third moment and vanish on diagonal indices. These estimates are expressed in terms of intrinsic (and easily computable) parameters associated with the random tensor $\boldsymbol{X}$ and the given coefficients $\langle θ_i:i\in [n]^d\rangle$, and they are optimal in various regimes. The key ingredient -- which is of independent interest -- is a combinatorial CLT for high-dimensional tensors which provides quantitative non-asymptotic normality under suitable conditions, of statistics of the form \[ \sum_{(i_1,\dots,i_d)\in [n]^d} \boldsymbolζ\big(i_1,\dots,i_d,π(i_1),\dots,π(i_d)\big) \] where $\boldsymbolζ\colon [n]^d\times [n]^d\to\mathbb{R}$ is a deterministic real tensor, and $π$ is a random permutation uniformly distributed on the symmetric group $\mathbb{S}_n$. Our results extend, in any dimension $d$, classical work of Bolthausen who covered the one-dimensional case, and more recent work of Barbour/Chen who treated the two-dimensional case.