Someone told me that in catan the possessor of the largest army card can move the knight each time it is their turn. Is this a common house or tournament rule? I could find no such statement in the rule book.
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.
No, this is not a rule.
There's no rule I could quote to prove this, but as you say, the rulebook doesn't mention it. The rulebook says when you CAN move the robber (when playing a KnightS/Soldier or when rolling a 7), but doesn't list all the different times that you CAN'T.
It must have just been a house rule; Catan has all sorts of house rules people play in their own groups.
No, Absolutely NOT.
The Longest Road and Largest Army are rewards for achievement, nothing more. Sure, the soldier/knight cards allow you move the robber, but only once when that card is played. You are allowed to continue playing soldier/knight cards after you've achieved largest army. In fact, it's not a bad idea because if you have played 3 and someone comes along and plays 4 then you lose your largest army. As long as you keep playing soldier/knight cards you can continue to move the robber, but never more than once per card.