This is a classic example of people looking at a document with a conclusion already in mind, and mistakenly assuming that the document they are looking at supports their preconceived conclusion solely because of some superficial characteristic which may not actually be relevant. In this case the logical fallacy is the assumption that if President Obama is keeping records secret those records must relate to his birth certificate or school records because you already believe those things to be falsifications which he would want to cover up.
This is an example of the Existential fallacy or Vacuous Truth fallacy which is often demonstrated in the classic False Antecedent Syllogism. In this case:
To those already inclined to belief this may sound good, but in fact there is no proven foundation for the first assertion and no actual causal link between the second premise and the conclusion. It's a compound logical error originating with that unproven assumption which the evidence introduced doesn't actually support despite all claims..
This problem of misreading documents which have some superficial elements that can be misconstrued by those inclined to look for nefarious meanings is remarkably common. It becomes even worse when the documents deal with legal terms or concepts which the readers are unfamiliar with, or when they use foreign phrases, usually in Latin, which can be easily mistranslated to produce nonsensical or self-serving results.
Entire pyramids of conspiracy theory are built on this kind of illogic, and their delusional adherents will cling to them relentlessly so that their entire delusional world view isn't brought crashing down by pointing out their foundational error.