Note that "combinatorial Hodge theory" can be in the context of fans (as in the work of [Karu][1]), Coxeter groups (as in the work of [Elias and Williamson][2]), or matroids. While there are certain common techniques and lemmas, there are essentially no general results in combinatorial Hodge theory. I will focus on the case of matroids.

In the setting of matroids, there are only a few papers which prove results that one could call combinatorial Hodge theory. The majority of papers in the area are studying various geometric models which are associated to matroids, and either applying known results to extend to non-realizable matroids, or using techniques that don't require combinatorial Hodge theory. 

In my opinion, the most important thing is to understand the geometry of various varieties which are related to matroids. A great source for this is Eric Katz's survey [Matroid theory for algebraic geometers][3], which is quite detailed. Unfortunately, it's from 2014, and a lot has happened since then. More up-to-date is Chris Eur's survey [Essence of independence: Hodge theory of matroids since June Huh][4], but it is not as detailed. Chapter 2 of [my thesis][5] also contains a survey of some results about the geometry.

It is not really necessary to understand the proofs of the combinatorial Hodge theory results in order to apply them, and some of the results have no exposition other than the original papers. If one wants to learn the proofs, I would recommend reading [A semi-small decomposition of the Chow ring of a matroid][6] by Braden-Huh-Matherne-Proudfoot-Wang and ignoring the discussion of the augmented wonderful variety/Chow ring. 

[1]:https://arxiv.org/abs/math/0112087
[2]:https://arxiv.org/abs/1212.0791
[3]:https://arxiv.org/abs/1409.3503
[4]:https://arxiv.org/abs/2211.05724
[5]:https://mattlarson2399.github.io/Papers/LarsonThesis2024.pdf
[6]:https://arxiv.org/abs/2002.03341