Pull files in the app's internal data dir

adb_pull.sh

#!/usr/bin/env bash

adb shell "run-as $1 cp '/data/data/$1/$2' '/sdcard/$3'"; adb pull "/sdcard/$3"; adb shell "rm '/sdcard/$3'"

PullADB.md

The app must be debuggable.

Another approach using adb backup is shown on http://blog.shvetsov.com/2013/02/access-android-app-data-without-root.html

Update

the command for Android 5.0+ to save /data/data/package.name/databases/file would be:

adb exec-out run-as package.name cat databases/file > file