Pedantry time: Walls of Great Court were in fact exposed to the weather - not covered in as James says, although much of the floor area of the court was indeed covered in bookstacks. A gap was left open to the elements around the edges. You can clearly see the effects of water on the facade surfaces down the years.
As for Foster's famous "Anstruther Claire" stone portico, my understanding is that this is the same geological bed as Portland where it surfaces in France. Which is not to say that the Museum wasn't diddled, only that the end result is the same.
There's a lot of xenophobia in the hysterical response from some quarters - what, French stone in our glorious imperial museum? (nobody ever talks about the smooth new Spanish limestone now cladding the Round Reading Room in the centre of the Court).
I have seen a sample of "real" Portland set against the new portico, and there is no discernible difference in texture or colour. Though I admit you'd have to have a very large sample - like, a complete second portico - to be absolutely sure.
Indeed, elsewhere in the Great Court, new "real" Portland has been used for patch repairs (and in a new band just beneath the new glass roof). This looks identical to Foster's supposed "wrong" stone in the portico. Because it is new.
Others have pointed out that, since any stone - particularly a sedimentary stone such as limestone - differs from layer to layer, there is no hope of ever getting an exact match anyway, since the areas being quarried now are far removed from those which sourced the original stone for the Museum.
So it's a real storm in a conservationist's teacup. But - personally, I'd like to know - what's Foster doing producing a classical stone portico anyway? As the original had long vanished, a good case could be made for a modernist equivalent, or none at all.
But Foster can (just) justify the huge expense and weight of the South Portico on good modernist functional grounds: it is the grandest of grand liftshafts. No, I don't think it's much of an excuse, either.
Removing my anorak of stone, I see that this thread is meant to be about the National Gallery extension in Dublin by Benson and Forsyth.