Not unless they want to take advantage of *all* the new features.
The new version of git will work fine with old repositories, both on the
"server" side and the "user" side. And people can use a lot of the new
features even if they do nothing at all.
But for the _specific_ case of having a clearly separated "local branch"
vs "remote branch" case, you do need to make that distinction clear when
you create the repository (unless you want to get really down and dirty
with the repo and just modify it yourself: certainly possible but
generally just not worth the effort since it's just easier to clone a new
one instead).
So it's really a matter of how you use it. Switching to a new version of
git on the "server side" (ie the shared repository operations) won't
really affect anything at all.
