Hi Steve,
Yes you can remove that but it is a module by module setting:
1 - Go the the module settings of the one that you want to remove the message from;
2 - Under "Module Settings" click "Advanced Settings";
3 - There is a checkbox called "Hide Admin Border";
4 - Just check it and save;
If you want to do that directly on the DB, perform the following update on the "TabModuleSettings" table:
update TabModuleSettings set SettingValue = 'True' where SettingName = 'hideadminborder'
Then remember to clear site cache as this information most likely is cached.
BTW, always backup before running an update query. It is the safer thing to do.
Cheers,
Aderson