For example, a work team (or academic department) will typically use mass emailing as their primary (or even sole) form of digital communication. But email is very poorly suited as a medium for large-group discussion. I see two major reasons for this: (1) the discussion is scattered across myriad individual emails, making it more difficult to refer back to past installments, and easier to lose replies. (2) Not everyone in the group will be interested in every such discussion, and so may not appreciate having their inbox cluttered by the constant stream of babble. A discussion board or 'forum' solves both these problems. So I think any such team (or department, etc.) that's likely to benefit from such discussions should ensure that they have a dedicated forum for this purpose.
A second example that jumps out at me is notification. Don't get me wrong: email is great for issuing one-off, instant notifications that may be of interest to others in one's team (department). But for serial or regular use, greater customization is called for. It's just plain rude for (say) the Italian Studies department to spam me every week about their upcoming public lectures. Rather than forcing such notification on students in other departments against our will, they should simply offer an RSS feed (or similar) which we may choose to subscribe to or not.* Similar lessons apply to trans-institutional announcements, e.g. conference announcements, "calls for papers", etc. It's completely backwards to rely on ad hoc email forwarding (all those "please distribute" emails sent to department secretaries) for this sort of thing. Better communications infrastructure should be put in place. For example, the 'PhilosophyCFP blog', if sufficiently well-implemented (I haven't looked closely), could render all those annoying CFP emails redundant.
* (Indeed, an optimally organized university would centralize such offerings, letting us pick and choose which departmental -- or even sub-field -- public notification lists we want to opt in or out of.)
These lessons may even apply to regular intra-departmental announcements: though I don't mind these as much, it wouldn't hurt to set things up so that recipients can pick and choose which regular departmental notifications they wish to receive. (But in this case, at least, the benefits might be modest enough as to not be worth the bother of setting up a better communications infrastructure.)
Of course, email overuse is a small crime in the grand scheme of things -- i.e. compared to the inexcusable overuse of snail mail. But don't get me started on the absurdity of requiring (e.g.) job applicants to transmit their information on dead trees...
P.S. Is there a standard administrative support position with this job description, i.e. an 'informational architect' to investigate ways the organization could streamline its communications? There should be.