How can I automatically save screenshots to Dropbox?

Faststone Capture can do some of this (including capturing only part of the screen). It allows you to specify a location to put the file and will even autosave (assuming you have Dropbox setup as a folder).

I love the program and use it daily to capture screenshots for training, documentation and for error codes.

Not freeware, but you can accomplish the same thing with SnagIt.

Create an All-In-One profile (allows for regions or entire screen) and have an autosave to the Public folder in DropBox (gives you direct sharing URL)

I find greenshot one of the best. It's light, fast, and has many options quick to access with right mouse button on the notification area icon.

Also, it's free and open source software.