In order to enable/disable the Demo Store Notice in Magento, you need to log into the Magento administration area and navigate to the System tab > Configuration
Select the Design section, click HTML Head and you will see a Yes/No drop-down menu for Display Demo Store Notice.
Enable/disable the demo store notice and click Save Config to save the change.
If you turn it on then you will see the message in front end like this
This message notice demo is standard magento notice UI, it can be customised by magento theme or extension. So that if you see somethings different from above result, it mean your magento notice message is customised by some third-party code.