Suppose I have a statement like "2+2=4". I believe the truth of that statement causes the truth of a statement like "2+2=4 OR 3=5". Have any philosophers or mathematicians developed the notion of a causal theory of truth? Such a theory should ideally require two statements to be relevant for there to be a causal connection between them.
Causal set theory, https://sites.google.com/site/lisaglaserphysics/research/causal-set-theory or https://en.wikipedia.org/wiki/Causal_sets or just google "causal sets", imposes a poset ordering on events corresponding to "causal connection" (your words). The logical analogy is just implication, and I'd think that's all there is to it. Although maybe substructural logical connectives better fit your intuition, which you don't really elaborate in sufficient detail for a sharp answer. For example https://arxiv.org/abs/gr-qc/0109053 which involves resource-aware linear implication in this case.
Yes, Whitehead and Russell did it. What you call "a causal theory of truth" is called logic in philosophy. Whitehead and Russell showed that logic and mathematics are one and the same.
That example given in your question is called the "principle of addition" in W&R's Principia Mathematica:
If q is true, then 'p or q' is true.
But I'm curious what lead you to this idea.