Here’s my take.
There are two aspects to this question: (1) Is the print journal format still required for reading purposes?; and (2) Is the print journal format still required to insure full access to journals given that many e-journals are under licenses (and are not owned by libraries) and digital preservation is still in its infancy?
It appears that the answer to (1) may finally be “no, for many users.” However, this may be contingent to some degree on the fact that many commercial e-journals are composed of article PDF files that allow users to print copies that replicate printed articles.
The answer to (2) is less clear, since continued access is contingent on periodic license negotiations and the changing business practices of publishers. Embargoes, ILL restrictions, incomplete back runs, and similar issues may give libraries pause. Very promising digital preservation efforts, such as LOCKSS and Portico still need to pass the test of time. Few libraries believe that publishers by themselves can be relied on to preserve e-journals (for one thing, publishers go out of business).
However, the reality for many libraries is that they have no choice but to dump print whenever possible for strictly economic reasons: print plus electronic is increasingly unaffordable for a variety of reasons.