I've seen some pretty strange use of technology.
For example, one place I worked was using Microsoft Message Queuing to send real time streaming VoIP data between servers.
I work as a consultant and many times the person responsible for these decisions is no longer at the company. The problem is that from the perspective of non-technical people it works just fine. The problem comes when needing to add functionality and fix bugs in systems like this.
If you are in a situation like this how would you build a case to spend time migrating to a more suitable technology?
You have to make a business case. In my experience, "This is an unsupported use" makes a great business case. Business people understand what it means to be unsupported by a vendor, and like to avoid that as much as the next person. Explain what risks or downsides the issue has to them (not to you).