This is a very bad idea, please don't try it because this will result in corrupted database. Here's why: while working, Evernote database manager creates so called transaction log files where information is written when database is being updated. These files are created in the same folder as the database file. Evernote expects that it's the only program that manages these files. Now imagine if Dropbox will shuffle them around and database manager will apply unexpected files -- immediate disaster!