I doubt it. A quick search on Worldcat reveals that there are multiple libraries around the country that have backissues of the Honolulu Advertiser on microfilm.
Are you telling me that someone went to the each of the dozens of libraries around the country and modified the microfilm? And that each and every one of them was doctored so well that it is undetectable?
Yeah, I suppose that's possible, just like it's possible Elvis is still alive or that the Feds staged the moon landing, but I it's a possibility so remote that no rational person would take it seriously.
No, pasting a new entry over old microfilm or microfiche isn't quite comparable to Elvis still being alive or the feds staging a moon landing. (Just to make it clear to you, curiosity, no one on FR that I know of endorses either of the latter two theories.)
The perps need not go to every library that maintains microfilm (or microfiche) from the Honolulu Avertiser from August 1961. (The number of such libraries is, BTW, probably much smaller than you think. Given Hawaii's relative geographic isolation from the mainland, interest in Hawaii newspaper archives is generally quite minimal elsewhere in the country - this controversy the exception to to the rule.) Access to only one such library or to the respective newspaper archives would suffice to produce an altered product whose image could then be produced on a pro-Obama web site. Furthermore, as others have pointed out, the absence of the birth announcement for the Nordyke twins in either of the posted newspaper Birth Announcement columns is consistent with alterations of the originals, assuming that all births filed with the Hawaii DOH over the week preceding the publishing of the announcements were published as a matter of policy.