Open Access - Harvard - Author Education
One way they can do this is to use one of the author's addenda available through the Scholar's Copyright Addendum Engine, which specifically requires publishers to acknowledge previously-granted licenses to a funder or to the author's employing institution.
The risk of not doing is so is that it may be that the author is making a misrepresentation to the publisher by purporting to transfer more rights than s/he can.
But, there's really nothing new here except that the licensee is the university instead of a funding agency. Government-funded researchers have been in this situation for decades. All government funding agencies (not just NIH) are required by OMB regulation to take a non-exclusive copyright license to any works created under a grant or cooperative agreement with a university. The government receives this license at the moment the work is created, just like under the Harvard license.
These government funded authors have been routinely signing copyright forms that appear to conflict with the government's previously-granted license. Legally, there are two options for characterizing what's been happening all these years. (1) These authors have routinely been breaching their contracts or have routinely been guilty of fraudulent inducement to contract by misrepresenting the rights they have; or (2) even though the text of the publisher-drafted copyright form says that the author represents that s/he is transferring all rights under copyright free from any licenses, that explicit text is actually subject to an implied term recognizing the previously-granted license.
Under the second interpretation, the boilerplate in the one-size-fits-all form is not controlling so long as the publisher is aware that the author is a government-funded researcher. The publisher should be aware of the government's license, and so by accepting the article, the publisher impliedly acknowledges that it is taking copyright subject to the government's license.
I prefer interpretation #2, and I think this is how a court would apply the law, but this is just an opinion. So we've been living in a world where authors should have been alerting their publishers to previously-granted licenses for a long time. Perhaps this new attention by faculty to their copyrights will lead them to address this practice as well.