I have had the same problem and some how resolved it this evening. 2 reinstalls, unpin/repin and location resets did nothing for me. But I did remove the last app I had installed, CNN (official app). Once removed I started getting toast alerts from WeatherLive again and the Live Tile is working properly.
Do you have CNN installed by chance?
I also removed USA Today and WP7 App List (rarely used). I doubt those 2 had any impact here as I have had them for some time.
This could also be perfect timing as I was getting an error stating "serverside error: Unable to save locations". Was that caused by something local (app conflict) or was something with the provider resolved around the time I dumped those apps?
I suspect something local because the MSN Stocks app didn't update the tile until I opened the app, WP Central seemed to be unaffected by this little hiccup as well.