Here're some observations that may help you track down the problem

 1. Magento's trying to include a template block with an empty string set as it's template. That's why you're seeing the warning. Look for any new (or old) setTemplate, or `template=""` in your code, or any place a template block is instantiated without setting a template
 2. Magento's trying to render the template system/cache/notifications.phtml — however, the fact its fallen back to the base design package to do this means Magento couldn't find this file in any adminhtml package/theme. This is a stock file that ships with the system, and may have been removed
 3. You say you're seeing it "everywhere", but the errors indicate it's an admin only problem. This might mean you're trying to run admin code from the front-end of vice versa (again, hard to remotely debug this sort of thing)