I don't think you have been paying attention to what recently transpired in the White House.
Through "an abundance of caution", the White House counsel advised and Obama agreed to have the oath of office administered a second time.
Now, why was that? I'll tell you why. Because any actions of Obama prior to taking the oath could be successfully challenged as being of no legal effect because Obama had not fulfilled the Constitutional requirements in order to exercise the powers of the Presidency.
This issue differs in no significant way whatever from the Constitutional requirement of being a natural-born citizen. If Obama is not a natural-born citizen, then any act he carries out in the White House can be successfully challenged. His signature on Congressional legislation would carry no more weight than yours or mine.
A reporter asked the White House spokesman whether Obama would be re-signing the executive orders that he signed prior to the second oath-taking. Though the spokesman seemed to be suggesting that it wasn't necessary, I predict that those orders will be re-signed before the week is out.
So he could properly take the oath. It has nothing to do with the birth certicate claims. Nobody in authority is taking any of this seriously. Not the court, and not Obama.