We have published a review article, years ago, and since then all of the authors left the institution. Now the institution has suddenly woken up and realized that the authors did not get permission before publishing the article. And the institute does not want its affiliation any more on the already published article. What to do now?
It sounds like you and your co-authors committed an error in publishing this paper against requirements imposed by your institution. The institution would probably be within its rights to insist that you withdraw the paper altogether, so asking you to remove the affiliation markings from the paper is probably not too bad a compromise (it's hard to know for certain without knowing the regulations and situation in detail).
As such, I would recommend getting in touch with the journal and requesting a correction to remove the affiliations, explaining the situation. The journal may or may not be willing to make the correction. If they are not, then you likely just need to leave the two institutions to battle it out and hope that you have a paper left when they are finished.