If you are talking about the Recent Files listing - it is not easy. You can delete the Preferences file but that will also delete other preferences.
I've been using this method:
Go to FileMaker Pro Preferences, and under the General tab, change the "Show recently opened files" to 1.
That wipes out the list of recently opened files, all bar one database.
Then you open one of your databases go back to the Preferences/General tab and change the number back to a number between 1 and 16.
But be careful here. If you change it back to, say, 16, it will also restore the old filenames which you were trying to delete. Very clunky.
You've got to change it to match the number of your existing dbs, and then open each of them one by one, and the recently opened list will now match your databases.
The actual filenames and path names are stored in the ~/Library/Preferences/FileMaker Preferences/FileMaker Pro 10.0 Prefs file. This can be copied from where it is, and opened to reveal the path names. Useful if you've managed to create duplicate databases and can't find out which is which in the Recently opened files list.
Or if you're brave, you can remove it to the desktop and it will completely remove the Recently opened files list, along with some other preferences. Quitting FMP and putting the preference file back in place will reverse the damage.
This answer solved a problem for me.