I have heard that the ban or cherem of Rabbenu Gershom has expired. Is this the case? If so, what are the ramifications vis–à–vis his enactments?
We must realize that Rabbeinu Gershom actually issued many Takanos (injunctions) and Cheramim (bans). Some were not accepted at all, some were restricted only to his generation, some were only accepted in the European Jewish communities, some had a specific time limitation on them, and others were accepted and embraced by all Jewish communities everywhere with no time restrictions. Similarly, some were enacted as a "fence" to ensure the keeping of Halacha, while others were enacted for the benefit of the social fabric of Jewish communities. Therefore, we can not compare the effectiveness and limitations of one of his Takanos to another.
For example, Rabbeinu Gershom's ban on bigamy was only accepted in the Jewish communities of Europe, and was originally only instituted until the end of 5000 years from the creation of the world (758 years ago). The only reason why we still continue this ban today is because the Rishonim who lived at the time that the ban was to expire decided to extend it indefinitely, as is stated by the Rema (Even HaEzer 1:10)....
The Cherem that we are discussing in our case, i.e. not to read another person's private information, was also instituted by Rabbeinu Gershom to be effective indefinitely, and was also accepted and embraced universally by all Jewish communities. Additionally, it was created as a "fence" so that people not transgress Torah prohibitions, i.e. the prohibitions of Loshon Hara (slander) and Rechilus (gossip) that would result from people knowing other's private business, and harm and damage that may befall people whose information becomes public knowledge, including informing (Mesira). This was evidently a prevalent problem in the times of Rabbeinu Gershom. Therefore, this Cherem must also be considered a "Cherem D'Oraysoh", and one must be stringent regarding it even in a case of doubt, as stated above.
The prohibition against polygamy was enacted by Rabbeinu Gershom, some thousand years ago. Some have suggested that the ban was only enacted until the end of the 5th thousand years (in the Jewish calendar), or the year 1239. Others, however, suggest that the ban was only made for one thousand years, and this might be the source from which one can derive that the ban will expire in the year 2037. Yet, the precise date of the ban is unknown, and some have written that although the thousand years are already up, the ban has been continually accepted as binding.
One way or another, the ban is mentioned in Shulchan Aruch (Even Ha’ezer 1) without any time limit, and it is unlikely that current the social structure will permit its disbanding anytime in the near future.