Except for one optional feature (noted below), the Dropbox app does not use your device's location data. However, because your photos may contain information about where they were taken, Dropbox (like other apps) needs special permission to access them.
Apple's privacy settings for photos are separate from location services, so the app will simply ask for permission to access your photos. In either case, the app will not be able to upload photos to your Dropbox unless you give the OK.
If you turn on the background uploading option in the Camera Upload settings, the app will ask for permission to access your device's location services. This will enable the app to finish large uploads without the need for you to open the app.
Here's how it works: Normally, photos and videos can upload while the Dropbox app is open and for only a few minutes after it is closed. But with background uploading turned on, the app will automatically resume unfinished uploads when it detects a significant change in your device's location. This feature is designed to make the Camera Upload feature work more efficiently and does not collect your location information.
When the background uploading option is set to On, you'll see the location services arrow appear at the top of the screen:
This icon usually means that an app (such as a mapping app) is using your device's GPS system (if applicable) to keep track of your precise location—and eating up a lot of battery in the process. However, Dropbox's background uploading feature uses a low-power version of Apple's location services. If you have a big backlog of photos to upload, you may notice that the app uses some power. But once that's done, uploading new photos should have little effect on battery life.
For information such as how to tell when an app is using location services, steps to reset or change your location settings, and to learn more about iOS location services, see Apple's Understanding Location Services support article.
For iOS developers or highly technical readers, check out Apple's Developer documentation to learn more about the API library Dropbox uses to access iPhone and iPad photo galleries.