Depends on the game. Not every Origin games requires origin to be running while you play the game. I own Dragon Age 1 and 2 on Origin, and I can just boot em up without it running. However games like Battlefield 3, and Mass Effect 3 require Origin to be running for some stupid reason, so those will run in the background.
Edit: By the way, thanks so much for the help earlier. This update just makes me so happy. i couldn't stand looking at the grey boxes for my non-steam games. I've literally bought steam games before even when I owned the retail copy because I hated the gray boxes...
I'm not really good at making pretty icons, if anyone has some good ones to use for the following games, please post. Final fantasy XIV, Starcraft II, Tera, and To the Moon. those are all to complete my collection.