Can the owner of the largest army move the robber every turn?
This is not an official rule; that is why you didn't find it in the rule book.

The reason this isn't a rule is because it would detrimental to the game experience. The player with the 'largest army' has already moved the Robber at least three times and has 2 victory points. Giving the player the power to move the Robber each turn, without a roll or playing a Knight card, allows them to squash the efforts of the other players. 

Thusly empowered, the player with the Largest Army could 'attack' the other players who are trying to gather resources. It would be hard to build the Longest Road, or upgrade their villages to Cities for Victory points, the robber keeps shutting down the resource tile you're depending on. From here, the optimal strategy would be to create and maintain the Largest Army, than hold down your opponents while racing for the other eight Victory Points.