I've had this issue for months now on my 920. I've un/reinstalled several times but it appears that this may be one of a handful of bugs in the live tile functionality in wp8, and for me it is only the weather apps that get hit by this bug. Some threads seem to suggest this stems from an issue with the periodicTask or resourceIntensiveTask implementation and has been mentioned several times on this and other forums before. Microsoft does seem to be aware of it, but I was hoping GDR2 would fix this issue, and since it's popping up on a 1020 it looks like it is still an issue.
You can try some of the following, but they haven't worked for me, but seem to have had previous success with others:
1. Unpin, uninstall, clear other storage and internet history/temp files, reinstall and repin
2. Standard shut down and reboot
3. Soft reset (hold power + volume down)
I haven't tried this yet, but its next on my list of tests:
4. Try uninstalling all apps that are in your background tasks list. Reinstall the one you are having trouble with and see if it updates after a power down cycle. Then add the rest back in one by one to see which is killing your apps tile.
I have not tried this, since you'll have to setup your phone from scratch again (contacts, apps, start screen, etc)
5. Hard reset - settings > factory reset
Take a look at these links, they may be helpful for you or at least detail the issue
http://social.msdn.microsoft.com/Fo...-api-background-agend-and-internet-connection
http://forums.windowscentral.com/windows-phone-7/190258-live-tile-issues.html
I'll post back if I can get my tiles working again, but maybe GDR3 will fix this...