Dropbox has an API that will (theoretically) let you do this, but there hasn't been a ton of people jumping up and implementing it yet. It'll be cool when it shows up.
I've used the dropbox API before to automatically upload photos in a dropbox folder to Flickr. It occurs on an interval (cron job every 2 mins). I'm sure you could do the same thing using FTP or a custom API that exists on your destination server.
They released a delta API recently. It's a bit of a pain to perform this task as you can't differentiate between a new file and a renamed file or a file that's been moved from one location to another.
In all cases you get a delete (if the file isn't new) and then a new file event.
I implemented it, and it kinda sucks for this sort of thing. The purpose seems to maintain local state to mirror the state on Dropbox. Not terribly interested in that...I just want to subscribe to specific events (webhooks, anyone?).
This was their solution to the frequently requested webhooks. It falls short. Way short.
ifttt.com has a dropbox-to-flickr recipe (http://ifttt.com/recipes/6804). Their dropbox channel provides 2 triggers, one for any new file in your public folder, and one specifically for new photos in your public folder (it doesn't say exactly what the definition of "photo" is, though).
I haven't tried it, so I don't know how gracefully it handles renames or moves.
