Counterfactuals in Pearl's sense are what allow particular models to be causal: to represent cause and effect under intervention, as opposed to mere correlation. This is an important part of how to build models that think like people[1].
Is it in particular the dot product (correlation) in MLPs that prevents them from inferring all causal structures in the data? So, instead of template matching of co-occurrences of features in the layer below, we (also) need to learn whether and how one feature causes the other?
Again, it's the lack of counterfactuals: the ability to intervene on a node and cut it off from its parents, then see what happens, and the ability to perform inferences over discrete spaces.
Are there any concrete attempts at transferring this concept to MLPs? E.g. by overriding the values of particular nodes/features by feedback connections?
No, because neural nets do not work that way, even when they output actions. Making things More Neural doesn't make them better, and AFAIK, not everything good can be made More Neural.
Are there works that expose this limitation of MLPs more formally?
>not everything good can be made More Neural.
Neural networks are universal function approximators, so you probably mean not everything good can be made with MLPs trained by gradient descent?
>It's the lack of [...] the ability to perform inferences over discrete spaces.
How would you judge the extent to which AlphaGo has learned to react to single discrete changes in the input. It seems that it learned very well to react very sharply to whether a single stone is placed at a strategically significant position.
[1] https://arxiv.org/abs/1604.00289