It isn't an app issue, it is an OS issue or a background task issue...sometimes, they hang. Ensure you only have background tasks enabled for the things you want to run in the background, reboot, and see if it gets better.
Amazing Weather is what I'm using. I figured it isn't the app but it's weird in that is does update the live tile on rare occasion. Also, the photo hub seems to stay current.
I still haven't had luck with Amazing Weather. If I launch the app it will update the live tile once. That's about it. Even leaving the app in the background doesn't work. I've gone through all the background apps and rebooted too.
It's not just Amazing Weather. Other apps seem to struggle to stay updated.
Every once in a while Weatherbug will update its live tile.