Sims3 games shown in Origin still need the disc to play

He/She is correct about being able to play without the disc through Origin.  I don't know if this is your problem, but a common mistake in this situation is that you must have the digital version of the game installed.  If you downloaded and installed the game using the disc first, its going to keep asking for the disc.  So you need to uninstall any games you installed with the disc.  Once the physical copies are removed from your computer, you can download the digital versions through Origin.  Just to be safe, make sure no disc is in your drive while you're doing this.  If this is done, then you should see the "play" option in Origin.  That's how I had to do it for my games.  Best of luck.    

edit:  Also, no you don't have be logged into Origin everytime to play.  I play my games without even opening up Origin.  I just click on the shortcut of the game as usual.