Update Oct 2018: Background image support has been added to this client.
One of the features of Gmail on Android is that you can configure it to access email from non-Gmail accounts. However it appears that when doing so, background images are not displayed when emails from non-Gmail accounts are opened. Thanks to Ann Marie Gingrich for bringing my attention to this issue.
I did a check myself and indeed all background images – table backgrounds as well as
background-image CSS – are not showing up in emails retrieved from non-Gmail accounts. Background images show up fine in the Gmail app if the email was sent to and retrieved from a Gmail email address (ie. gmail.com).
This problem will become more acute as Google has dropped the native Android mail client in favor of the Gmail mobile app for their newest operating system starting from 5.0 (Lollipop). Users on Android devices who need to access their work email would have to use the Gmail app or a 3rd party mail app.
Update: This setup on the Gmail App is now referred to as the Gmail Android app with Non-Google Accounts (GANGA).