Great question. Likewise is a private, nonprofit college that serves the needs of indigent prison inmates. Our model requires a large number of volunteers who help faculty and staff handle the various emerging needs. So, our first line of defense against undesirable tasks remaining undone relies on the servant leader disposition of volunteers.
Likewise cultivates 4 volunteer channels (from Luke 10:25–37's "love the Lord your God will all your heart, soul, strength, and mind"). The "strength" group invites "non-academic" talent contribution (e.g.-transcription services, since our students cannot access computers and must submit handwritten assignments). Faculty may volunteer to fulfill this need. When they do, they earn "development points" (from developing their volunteer/non-direct-compensation disposition). These development points increase the faculty members' next contract according to formula (up to doubling regular direct compensation). Likewise sets aside the development funds whether faculty determines to increase their value to the organization or not. Of course, if faculty members fail to handle the opportunity to their benefit; and volunteers fail to materialize; then the service falls within general staffing expenses — which keeps the saved funds from being applied into the general faculty support endowment (since staffing needs for the activity consume resources). We recognized the need for exact transcriptions during the prototyping of 'TR-1000: The Books of Luke' and placed the need into the existing budget process so that either "strengths" volunteers (or faculty acting as volunteers for future benefit) or staff might fulfill that need.
To summarize using this example, if faculty wants to voluntarily fulfill transcription needs, they thereby increase their future value to the organization while helping to control staffing expenses. Otherwise, other volunteers usually step in and handle the 'unwanted' activity. If an undesirable and essential activity falls through these cracks, then I raise the tension to the staff circle's governance meeting and we adjust (preferred) or create (last resort) a role and its compensation to cover the unfulfilled non-academic task. However, this is a last resort as it removes an opportunity for faculty development.
One final thought, if an activity is truly unnecessary, then why are we doing it? More likely, we need to show how the activity connects to the overall mission. Partners who value the mission ought to understand the value of every necessary activity and that their commitment to the mission naturally requires an interest in activities even as mundane as checking the mail. Can they imagine fulfilling the organization's mission without an awareness of whatever might be sitting in the mailbox!!??!! So, unless your example is purely hypothetical (and even if it is), an emphasis on connecting even the most mundane acts to the mission needs to happen--and regularly. If it's work that's essential to the organization's purpose, then figure out how to elevate the work's importance. If no one in the organization values the service that the work does to the organization, maybe it's time to hire someone who will value the organization through the work...