Warning: include(File.php) failed in Autoload.php on line 93

the Autoload tries to require the file for a class that isn't found in the code base yet. Since it is requiring File.php it stands to reason that somewhere in the code the class File() is called which doesn't exist yet in your codebase.

If you track down the spot where that class is instantiated you will have fixed the issue.

For follow-up, the mageDebugBacktrace() provided the following result:

[1] :
[2] :
[3] /var/public_html/domain.com/live/app/code/core/Mage/Core/Model/Cache.php:199
[4] /var/public_html/domain.com/live/app/code/core/Mage/Core/Model/Cache.php:119
[5] /var/public_html/domain.com/live/app/code/core/Mage/Core/Model/Config.php:1348
[6] /var/public_html/domain.com/live/app/Mage.php:462
[7] /var/public_html/domain.com/live/app/code/core/Mage/Core/Model/App.php:401
[8] /var/public_html/domain.com/live/app/code/core/Mage/Core/Model/App.php:295
[9] /var/public_html/domain.com/live/app/code/core/Mage/Core/Model/App.php:337
[10] /var/public_html/domain.com/live/app/Mage.php:683
[11] /var/public_html/domain.com/live/index.php:71

Looking at the Cache.php file, I can see that the 'File' part is the default backend cache type used by Magento, if none is specified. I changed my app/etc/local.xml file to use 'File', note upper case 'F':

File app/etc/local.xml:

<cache>
    <backend><![CDATA[File]]></backend>
    ...
</cache>

Hope that helps someone out.

Just encountered a weird Warning: include(somefile.php) failed in Autoload.php on line 93 after pulling work on a shared test environment and someone else enabled the cache.

Not this exact case, because it's mentioned:

Ensured that the caches are emptied and indexing has been run several times.

So, always remember to check and disable/clear the cache if working on a shared environment, before wasting hours debugging!