Here's an Apple discussion thread on the issue.
And here's my reply in that thread. The latter part is of general interest, and addresses a common misconception about multitasking in iOS.
BTW, I just deleted and recreated that reminder, so I'll check it tomorrow.I too found that my one location-based reminder stopped ringing after updating to iOS 7.
Outlook is not an issue as the only e-mail account I have is .mac (iCloud).
I have checked all settings, but have not deleted and recreated the reminder yet. So unless there's an issue with legacy location-based reminders going haywire, while newly created ones work, this is an iOs bug.
As to the multitasking bar issue, I think there's a fundamental misunderstanding. The list shown when you double click the Home button is not a list of apps running in the background. It's a list of recently used apps.
iOS dynamically quits apps that aren't in current use when it needs the memory. There's no way for a user to know if Reminders is "in the background" as opposed to quit. (Except, of course, if you force quit it.)
Furthermore, it makes no sense to quit apps from the multitasking list unless they're failing in some way. (Or eating battery or data usage when not in use, which I would call a failure.) iOS is designed to take care of that for you.