drag on means, to go or pass slowly. For example,
The meeting dragged on.
drag out means, to get, pull or take out something that might not be easy to lift or do. For example,
Let's drag out the winter tires. They're big, bulky, and heavy.
I don't want to drag out our family photos. It's too much of a bother.
Drag out your friend tonight.(s/he might not want to go out, so you might have to push/pull her/him by using force or persuasion).