When you use Dropbox to sync your Papers library across multiple devices, a lock file notification appears on your computer ever once in a while. There is no setting in Papers to switch this off, as the update is coming from Dropbox rather than Papers. However you can switch off those notifications by clicking on the Dropbox icon on your menu bar, click the gear symbol and select Preferences. Finally, on the General tab, untick the box for "Show desktop notification".

The Lock file is required by Papers to tell other copies of Papers connected to the database that the database is in use. It's a very small file, so it won't be taking up much space. It doesn't mean that the app is syncing, but it does stop anything else from accessing the database at the same time (which would cause corruption).