Yeah, it's hard to say. It could just be that he wasn't paying attention and forgot, but since Melania was holding them you'd think she might have brought his attention to it. Or there could have been something more sinister. There's credible information (posted by Mark) that Trump had converted to Judaism, and so he'd not want to put his hand on a Bible that included a "New Testament".
It also reminds me a bit of the "Astronauts Gone Wild" scenes (by Sibrel) where they refused to swear on a Bible they'd been to the moon, and Buzz actually punched Sibrel (that was actually funny and Sibrel kindof had it coming), but I did wonder about it. So, most of the astronauts are docuмented Freemasons, so are they Masons who still believe in the Bible sincerely, or they believe but reject it and yet are worried that if they lie on a Bible they'd be punished by God, or whether they don't want to swear on it because they despite the Bible. Not sure why they'd refuse to just put their hand on it and lie.