Same issue here, I have found this is mainly in part of  the total CPU usage as illustrator subsist being a memory hog by nature so keep in mind size total of open files and how many files are open at once. 

I have the same problem using Windows 7 / 64-bit. The problem has existed since the last update from Adobe. I was hoping the next update would fix it. But, no more updates ever rolled out.

It's not a permissions problem with or corruption in preferences. Preference changes apply correctly when modified and are retained after re-launching Illustrator.

Most of the products in the Creative Suite have major efficiency flaws. So, Illustrator being a CPU hog at times when it shouldn't even be active comes as no surprise. I'm not bad mouthing Adobe. It's just the way the software industry seems to work for most.

Thanks for updating, "USGrant7".

I could not find any information on the "unverified"/"verified" status or the "verify button" you spoke of. I presume you were referring to functionality found only within "Windows Process Explorer" (for the sake of most other readers, Windows Process Explorer is a third-party utility and not part of Windows).

I do not have Process Explorer. And, the problem is not bad enough (or hasn't gotten annoying enough yet) to warrant me installing an application for such a limited use. However, if I cannot find a more conventional fix, I may eventually give it a try.

Update:

I did find another clue to the bug. I thoroughly searched Adobe's online fix-it-yourself information, trying everything that seemed applicable and reasonable. One thing finally had a direct effect, though not an appealing permanent solution for me personally. Solution 8 (see link below) instructs to try opening Illustrator in a new user account. Within the new user account, the problem does not ocur.

http://helpx.adobe.com/x-productkb/global/troubleshoot-system-errors-f reezes-windows.html#main_new_user_account

For a geek like myself, I have a lot of time invested in setting up and tweaking my user account to suit my taste. So, given a new clue to find the actual problem, I kept digging. Adobe's solution presumed to diagnose the cause stating, "If the problem doesn't recur, your standard user account lacks access rights to files and folders the Adobe application requires." I tried granting myself full control over everything in any Adobe folder I could find; in both Program Files folders, also in ProgramData and my user account folder. I eventually granted myself full control over everything in my user account folder. None of these attempts made any difference.

A closer look at Adobe's statement diagnosing the cause reveals an incredibly broad blanket specification, "...files and folders the Adobe application requires." That could mean anything, not just Adobe files or folders. I mused over what specifically Illustrator could possibly be trying to unsuccessfully access, only when closing, and that no other application in the Creative Suite would access. Other than a possible corrupted registry entries for Illustrator, "I got nothing."

Finally, A FIX !!!!!!!!

Here's the skinny:

• Open the TeamViewer window.
• From the menu bar, select "Extras" and "Options".
• Select the "Advanced" tab.
• Click the "Show advanced options" button and then the "Configure..." button (for "QuickConnect button").
• Add "illustrator.exe" to disable the QuickConnect button for it.
• You may also want to add "flash.exe" if you have professional installed, which is reported to have the same problem.

See my post of "Illustrator CS5 crashing on exit" for more information on this problem.

Good luck and happy illustrating without crashing every single time you exit Illustrator.