Warner Bros. announced that Batman Arkham Origins' servers are shutting down on December 4th, 2016. It means that the game's multiplayer side will no longer be playable. However, this will have no effect on it's single player side, which will stay same as before.
Batman: Arkham Origins had trophies and/or achievements, including online ones. But, as of now, what'll happen to those multiplayer achievements/trophies after this event is unknown.
This is the official statement, "As of December 4, 2016, the online services portion of Batman: Arkham Origins will be retired," they added, "We thank those that have joined us to battle over the last 3 years. The single-player campaign may still be played and enjoyed offline."