Welcome to Magento.SE!
I have thoroughly reviewed the codebase and the demo store at http://enterprise-demo.user.magentotrial.com/checkout/onepage/ and could not find evidence of a 'Define Order Date' field at all in Magento 1.9CE/1.14EE, or any prior version.
Some troubleshooting tips:
Disable all local modules by editing the app/etc/local.xml
file and changing the node to <disable_local_modules>true</disable_local_modules>
Revert to the default theme (base/default) and see if the issue persists
Install a from-scratch version of 1.9CE with the demo data and see if the field continues to display.
The only "date" field that shows by default in billing information step of checkout is the Date of Birth field. These are configurable, too, by editing System > Configuration > Customer Configuration > Address Templates
Otherwise please review the standard debug procedure here: https://www.sonassi.com/knowledge-base/magento-debug-process/