It seems Command & Conquer 3 Tiberium Wars has a major issue with running as a standard user. Symptoms include standard users not being able to run the application, with the system reporting that it does not have permissions.
In my experience this is due to the update application not setting the proper permissions on files after modifying them, it basically removes the users group from the CNC3.exe and other files.
To resolve this issue navigate your way to the Electronic Arts folder. Right click on the Command & Conquer 3 folder and click Properties.
That will open up the Properties window, you'll want to click on the Security tab and then click on the Advanced button.
You'll then need to click on the Edit button near the bottom (you'll notice its got a little shield icon meaning you'll need to elevate at this step), you'll then be presented with this window, which at the bottom has a check box next to 'Replace all existing inheritable permissions on all descendants with inheritable permissions from this object' you'll need to check that box, and then OK your way back out.
It will then go and set the proper permissions on all the objects inside the Command & Conquer 3 folder. That should resolve the issue. However unless EA fix their update application it could well happen again when they next release a patch.
Other than this oversight the application otherwise runs fine as standard user, and installs fine from a standard users' desktop while elevating using UAC. So it is only a relatively minor fix for EA to apply at their end.