SouthpawJosh: I can't be sure that my solution will work on every phone, but if I were you I'd try some other live tile; WeatherFlow can be stuck with a completely different problem (not necessarily their bug, maybe some other rare fault of the OS). If you already have other third-party live tiles that work then your problem is definitely with WeatherFlow and I would insist with the developer to find a solution.
What I can tell you about WeatherFlow is that they did something around the usual mechanism of the background task; I know this 'cause my WeatherFlow live tile updates exactly every hour and, as a developer, I'm sure that the OS doesn't offer such an option (if you use background tasks in your app you will be called every half hour); so, it's clear they've done something clever to limit the amount of data used by their app (in fact frequent updates on weather forecast are useless), that is: they do something in their background task only half the times they're called, thus they check for something before executing the update...and maybe there's a problem there. To do this they have to save their state and maybe they don't save anything when your storage has less than 10% free space (I don't know, I'm just imagining a scenario).
Anyways, my point is: one live tile that's not working is probably a different problem than the one we're having here. Also: my live tiles (WeatherFlow included) stopped completely (for a week) until I found that solution, it didn't happen that they worked for a brief moment and then went dark again.
F.