Syncing .apps via dropbox/resilio sync
Firstly, DropBox, Resilio sync and other such systems do have limitations about the types of files that they can sync. They may not sync certain file types, metadata, files with particular character sequences, and so on. You should read up on these, as this may affect syncing apps.

Some apps can have components in /Library, and/or the User's Library. These components may get updated by an installer package.
If an app gets updated through syncing, but all these other components are not updated (which would have happened in an updater), then you might run into trouble if a newly synced app tries to read support files from an older version. (It's also unwise to sync severl areas in the Library folders, as these can be machine specific.)

My gut reaction is that it might cause unforeseen problems, and given that most apps either update themselves or use the Mac App Store to do so, (and are otherwise static) there's not really any compelling need to sync them.