Base Breaker: While O'Reilly himself identifies as a political independent, he's become a much more polarizing figure among conservatives in recent years, most likely in no small part due to the fact that he hosts a show on Fox News, a channel that's generally more on the conservative side of the spectrum. His support for gun control, LGBT rights, and some degree of amnesty have made more ideological grassroots conservatives hostile towards him, labeling him a RINO for his breaking away from more traditional conservative stances. Generally, the biggest problem conservatives seem to have with Bill is that whenever a liberal/progressive cause finally goes mainstream and has the support of a majority of Americans, Bill O'Reilly will always get with the program and either drop his opposition or soften his stance towards said liberal cause.
Critical Research Failure: The infamous interview with David Silverman and follow up vlog, where Bill tried to argue that, because science can't explain the tides (it can) and moon (ditto) then God must exist to be able to do it. Even if his beliefs about scientific understanding were correct, the idea that unexplained phenomena can only be filled up by God - the so-called god-of-the-gaps argument - is one that actual religious theologians have long since abandoned as discredited.
Dietrich Bonhoeffer: How wrong it is to use God as a stop-gap for the incompleteness of our knowledge. If in fact the frontiers of knowledge are being pushed further and further back...then God is being pushed back with them, and is therefore continually in retreat...God wants us to realize his presence, not in unsolved problems but in those that are solved.