Source for this claim: "In high dimensions, all data are multicollinear." In the encyclopedia of statistics article on the curse of dimensionality, one reads:

Scott and Wand [5] show that high dimensionality increases the
probability that a data set of fixed size n lies close to some proper
subspace.

But looking in the referenced paper, I don't see this problem explicitly treated in the paper (the "show" part of the sentence above). In any case, I would have liked to know of other papers that study this.
Edit:
Intuitively, the LLN applies to the rows $\{x_{ij}\}_{j=1}^p=\pmb x_i\in\mathbb{R}^p$ of my $n\times p,n>p$ data matrix $X$, we have that for any positive number $\epsilon$, $1\leqslant i\leqslant n$,
$$\lim_{p\to\infty}\Pr\!\left(\,|\text{ave}_{j=1}^px_{ij}-\mu_i| > \varepsilon\,\right) = 0.$$
If the rows of $X$ are identically distributed, then $\mu_i=\mu\;\forall i$ so that in this sense the data would be lying closer and closer to an hyperplane. Is this the intuition behind this claim?

*

*Scott, D. W. and Wand, M. P. (1991). Feasibility of multivariate density estimates.
Biometrika, 78, 197–205.

 A: This response is likely not the explicit treatment requested. This question 
reminded me of David Scott's comments at the first joint ASA/NSF conference on Massive Data when he said, "The only thing massive about massive data is the massive redundancy." In terms of analysis of high dimensional data, his recommendation was to focus on the mode as the best measure of central tendency and he's since published papers on approaches to mode clustering. But his views are largely unknown and, for all intents and purposes, lost on today's audience of analysts.
More common is the opposite view about the relation between collinearity and data size. For instance, in Gujarati's widely used textbook on econometrics he describes multicollinearity as a small data problem where a theoretical set of predictors "are not linearly related in the population" but can be in the finite data sample at hand. In this view and since all data is finite -- even at the petabyte level -- the potential for collinearity rapidly explodes in high dimensions.
Finally and related to this is the intuition driven by the assumption that predictors be iid, at least in the theoretical, asymptotic, infinite limit. In the reality of applied analysis, however, data is rarely iid and dependence structures -- both linear and nonlinear -- where everything is endogenous and everything is correlated are rife. 
