But you can sync to your Dropbox account. This is quite different from having your Database file stored in your Dropbox folder, and is fully supported by DEVONthink… and has the same effect as you want…a remote backup.
Start by going to Sync in Preferences, and add your Dropbox Account in Locations.
You also need to go Selective Sync in Dropbox and exclude Devonthink in the “Apps” section to avoid it being downloaded to your Dropbox Folder on the computer.
If you ever needed to, you could recover the database from the Dropbox account, by going to “Import Database” in the Sync prefs, and selecting your Dropbox account as the source.
There is some guidance on this in one of the tutorials, and it might be worth buying the Take Control of Devonthink book.
EDIT you might want to look at this thread for more discussion of this. Although syncing to the Dropbox account is fully supported, and can be used as a means of restoring a database, there is a lot of resistance to regarding this a backup, even though it is. The mantra about not having your databases in your Dropbox folder is so strong that using the supported Dropbox method seems to be frowned on.