I appreciate the advice, but perhaps I didn't make it clear that the login requests were happening on my phone, not on my Macbook Pro. As such, there was no exclamation point or other alert besides the annoying dialog (in which iOS's various APIs do not, annoyingly, include a "Never ask again, and by the way, update your feed links!" option).
The problem appears to be solved. Apparently simply killing the iTunes Store task still leaves search and download processes resident; I reopened iTunes Store, cleared the search and entered another, and then canceled two downloads that were still valiantly trying to finish.
It's been about 36 happy iPhone using hours since I've had a mevio.com login prompt. Even so, in a perfect world:
1) mevio's servers would finally be taken down completely, disabling the login prompt;
2) podcasters who still have iTunes and other feed links pointing to mevio would receive a mild electric shock every time the dead links caused user issues;
3) Miley Cyrus would be a mime. (This last one is not germane to the topic, but in my opinion, it would bring us closer to a perfect world.)