This is not how the Dropbox API works at this moment.
There is no mechanism to push updates to your application. Only the desktop client gets async updates. It will be a glorious day when such an API exists.
But don't hold your breath, it's been a topic on their forums for years.
Push events to the web app isn't important to me. Syncronization with a web app every few minutes would work well, and you could login to push a button if you needed the changes pulled immediately.
That's the way i implemented it in an iphone app i recently made: a 'sync now' button to pull anything new and push any updates to the server. Works quite well - I recently deleted my app (iphone died) and reinstalled it, one 'sync now' later and all my data was back.
I'm a big fan of using dropbox like this, and am hoping to explore this approach in a lot of productivity apps i plan to make in the future. I like the idea that the user can 'see' their data in a readable format (json,csv,yaml - i still haven't decided which i prefer!) in their dropbox\apps\MyApp folder.
It's like icloud but you feel in control, rather than 'my data is out there somewhere but i can't see it'.
My users would not be this excited if a "sync now" button on the web needed to be pushed. The point of Dropbox integration is to upload photos without leaving Lightroom/Aperture/etc. and not having to install a plugin: http://pgunupudi.blogspot.com/2009/09/dropbox-support-for-ou...
There is no mechanism to push updates to your application. Only the desktop client gets async updates. It will be a glorious day when such an API exists.
But don't hold your breath, it's been a topic on their forums for years.