As long as it takes to produce a forged AUTHENTIC birth certificate that is beyond reproach.
Id say that it would be nearly impossible, as too much time has gone by. Paper weight, thickness and chemical composition would need to be taken into account, along with moisture and acid content. The way paper is made has changed a great deal since then, So, unless someone has an authentic unused sheet from that period, it would be nearly impossible. The same is true with ink.
Then it would need to be artificially aged.
Far too much work to forge an old document that will get heavily scrutinized since it was printed. The best Obama can hope for is a record-destroying fire, or more likely ,a past record destroying fire to blame it on. Just calling it "Lost" or a "well timed catastrophe" would just add fuel to the fire.
But we need to wait -- there is still no order, And I suspect there wont be one for at least a month and a half.