Are you saying ONLY your weather tiles do not update or all your 3rd party tiles that are pinned don't update? Usually when you have tile problems that are not specific app related, it tends to be all of them. (Intermittent, not at all, etc.)
Check out the comments/suggestions by Mr. Brown and myself in this thread:
Live tiles dead
Basically it's most likely too many background agents running. (Some apps can have agents running and you may not see them in the 'background tasks' settings screen) I'd advise opening all your apps and checking their respective settings to disable Live Tiles, and other background agent/tasks there on top of blocking unwanted apps in the background tasks settings. Also keep a count of how many apps have background tasks running. More than 15 seems to be the spot where people start having troubles. Good luck.