Dropbox to continue backing up images until it completes
This happens because of a restriction put on all apps performing background tasks, specifically those fetching data. For example, this happens with the Spotify-app as well when it downloads songs for Offline Mode.

There isn't really any way to bypass these restrictions, at least not on a non-jailbroken device. It might be impractical, but the only option is opening the app every time it has to pause/terminate it's background tasks. You can of course also keep the app open – that will allow for the tasks to be completed with no practical time limit.

If you have jailbroken your device, you can download an app enabling what's commonly called "true" multitasking enabling apps, such as InfinityTask, which disables many of Apple's multitasking-restrictions.