Armor is fairly important for tanks in league, as it lowers the physical damage they take from enemies.
Armor absorbs a certain percentage of incoming damage to the champion, it is not on a 1:1 scale. For example having 100 armor does not block 100% of incoming damage.
Armor also does not scale up at the same pace throughout the game, buying 15 armor at the beginning of the game when you have 40 armor has a greater impact than buying 15 armor at the end, when you already have over 100 armor.
Finally, different items have a different cost-effectiveness for armor values:
Cloth Armor costs 300 gold and gives 15 armor, that is 20 armor per gold
Chain Vest costs 720 gold and gives 40 armor, that is 18 armor per gold
So taking into account the way armor scales down with the amount of current armor your champion has, and the cost effectiveness of different items that give armor, I'd like to know when is it economically right to just stop buying armor. Or
What is the armor value at the point where buying X armor will only increase damage absorption by 1% (or less), and what is the minimal gold value to get there?
For example, I played Garen on Twisted Tree Line and started with 40 armor, which said gave me a 29% reduction to incoming damage.
I bought a Chain Vest right off, giving me 82 armor (40 + 2 additional from my masteries), with 45% damage reduction
I proceeded to record throughout the game:
127 armor = 56% reduction
170 armor = 63% reduction
198 armor = 66% reduction
As you can see, the difference of 30 armour in late game only bumped it from 63% to 66%, but adding 40 armour earlier bumped it from 29% to 45%
Best Answer
So, here's the thing about armor in League of Legends -- the % reduction in the tooltip is not very helpful in realizing what armor actually does.
To be more specific, I direct you to my answer here. (I even use Garen in my examples!)
The short version is that you can think of each point of armor as adding a finite amount to the number of seconds you live in combat. The exact number of seconds added per point of armor varies (due to Health, mostly), but the important thing to note is that each point of armor adds the same amount. Which means we can answer your original question as follows:
Every additional point of armor provides an identical effect.
There is no "most cost-effective", because neither "cost" nor "effectiveness" vary (i.e. a chain vest always adds 40 armor; 40 armor always increases your lifespan by N seconds for identical health values).
So, you're looking at the problem the wrong way. You shouldn't be buying armor until you reach a specific % damage reduction -- you should be buying armor until you survive long enough to kill them before they kill you!
(You might be interested in maximizing your armor / gold spent... but that's beyond the scope of this answer.)