How can I remove Live Tile updates in Windows 8?

You can disable live update for each tile by right-clicking it and choosing "Turn live tile off":

In the interface, you do not have an option to change a live tile’s update frequency. There might be some registry hack to do it.

An easier way would be: To turn off individual tiles, right-click the tile on the Metro startpage, and select the turn live tile off option on the bottom toolbar.

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If you don't want to do this on a per-app basis, you can disable Live Tile notifications, Toast messages, and communications with the Windows Push Notification Service in the following Local Group Policy branch:

User Configuration\Administrative Templates\Start Menu and Task Bar\Notifications