Was there ever a case where one has to prove access to materials cited? One can obtain reference materials from various sources. If not of open access, it could be either bought from a publisher or downloaded with an institutional license or even borrowed from a colleague.

Just out of curiosity, was there ever a situation where authors must prove their access to the articles cited in their paper to the editor/publisher/reviewer?

I agree that some of the references need not necessarily require licenced access for the sake of referencing such as pointing to datasets. However, my concern is about original research articles from which you cite a piece of information in your paper.
I could interpret your question two different ways: (a) proving that you have read the article (because otherwise your research is academically dubious), or (b) proving that you were legally entitled to read the article (because otherwise your research is legally dubious).

If the material is published, then you do not need a license to read it, you only need a license if you want to copy it. It is perfectly legal to read it by looking over someone's shoulder on the bus or in the library. Therefore, you do not need to prove that you were legally entitled to read it.

On the other hand, if you don't have access to a copy that you can go back to any time, you are going to have trouble responding to any queries from referees that suggest you didn't read it carefully enough...

As for unpublished material, such as private letters, I think it would be prudent to refrain from citing the material unless you have legitimate access to it.