I have always thought that you would need root access to pull an SQLite file from a non-rooted Android device. Turns out I thought wrong! Here’s how you do it:

$ adb -d shell
$ run-as your.package.name
$ cat /data/data/your.package.name/databases/yourdatabasename  >/sdcard/yourdatabasename

This will copy your app’s SQLite db file to the SD card’s root directory. From there, you can copy the file to your computer any way you like.

Props to this SO answer!