An EA rep tells Joystiq that Live free members should go to the "Extras" menu and input the code in the "redeem code" section. This will unlock the single-player Zealot Suit and Zealot Force Gun without the need for a Gold membership. The problem with going through the Online Pass screen is that it's used for multiplayer, which you can't access without a Gold membership. This isn't the first (and likely not the last) time that DLC insert cards have caused confusion.
[Thanks, Dayton W.]