I have a few more questions:
1) where did you get your drivers from? the default version on 14.04 is 331.38
2) is this the latest version of CrossOver downloaded from this site?
3) can you provide the output of running /opt/cxoffice/bin/cxdiag?
(you can either enclose the result in [code][/code] tags or just put it up on pastebin and link it here)
Update: I'm actually wondering if this is a possible problem. Current CrossOver is based on wine 1.7.4 with some added "fixes" for various programs but according to this git page, at that time, the GTX 880m was not in wine's card "database". It isn't the the database in git head either, which is unfortunate.
http://source.winehq.org/git/wine.git/blob/18556fd521d27d298d22db5eb48e192a2c634e59:/dlls/wined3d/wined3d_private.h
There are two registry keys that one can set and maybe wine will feed the correct(ish) info to the game.
First make sure that the Battle.net Client is closed.
Open a text editor and copy the text from here: http://pastebin.com/az75mYgE. Name it "fix_nvidia.reg". Now open Crossver, go to Tools -> Manage Bottles (you may need to hover over the top bar in Unity to reveal the menu).
With the Bottle Manager opened, select the bottle where Diablo 3 is installed, click on the run command button and type regedit. This will open the registry editor and now go to Registry -> Import registry file and find the file you created above. Now you should have a key under HKEY_CURRENT_USER\Software\Wine called Direct3D with two values VideoPciDeviceID and VideoPciVendorID.
Close the registry editor. With the same Diablo 3 bottle selected in the Bottle Manager, go to the Control Panel tab, choose Simulate Reboot option and then the Launch Selected Item button.
Try running the script you created earlier again and see if the game works now.
I would like you to post, either here or on pastebin, the result of this command: lspci -vnn | grep VGA -A 12