Major premise: Logical arguments require demonstrable facts.
Minor premise: Beliefs are not demonstrable facts.
Conclusion: Logical arguments cannot be built on beliefs.
The main thing that bothers me about what the fellow said is that it seems to lack any factual content itself. It seems subject to the same flaws of old-school positivism. Is it an empirical fact that you need facts to make logical conclusions? How can I observe that statement? Or is the stricture itself just a belief about logic? Are facts, in fact, properly objects of logical construction?
The first thing to sort out is what the sage means by "being logical". As any student of logic knows, there can be perfectly valid arguments built upon unreal (i.e., nonfactual) premises, which makes them unsound. For example, "All quagborts are pleth. Merlin is a quagbort. Therefore, Merlin is pleth." Nothing in this (modus ponens) syllogism is based in fact––Merlin and quagborts are fictional, and pleth is an unreal attribute––yet its logical form is entirely valid. Valid, but unsound.
So, what the Youtube sage should be willing to grant in the first place is that a theist could construct a valid logical argument about God, yet only be able to find fault with its soundness based on a squabble about the premises (or, facts) employed in the argument. This is why I am certain an unstated premise of the sage's syllogism is that 'facts' refers to empirical, everyday, uncontested facts, like Kareem Abdul Jabar is taller than Spud Webb. By insinuating this premise into the argument, the sage tacitly canvasses all "rational" people to his side, since only "crazy" religious believe in things no one can "prove". Real facts are just obvious to anyone with open eyes, right? Right? (And the foundationalists went wild with glee!)
Unfortunately, it seems to me that as soon as you lower the argument to one about "whose facts" and "which facts", you've entered a field of discourse much more diaphanous and vast than the tidy, hermetic formalism of logic per se. Is there a purely logical way to parse facts, so that we know whether they are "factual premises" in an argument? What worries me about his position on logic is the minor premise, namely, that beliefs are not facts. It seems impossible to see that premise as anything more that just that––a premise, a belief, about the nature of belief. The minor premise, then, is a belief, built into the fabric of his logical argument, about the uselessness of belief in logical arguments! The basic problem is that his "afactual" belief about the factuality of logic is integral to his critical logic about the illogicality of beliefs.
The larger problem I have with his confusing bifurcation between facts and beliefs is that it ignores the more fundamental role of beliefs, as the girders of a total interpretive matrix, in establishing, or even recognizing, alleged data as facts. Behind any ordinary application of facts there lies an immense field of prior beliefs (viz., about causal order, object permanence, cognitive clarity, temporal succession, etc.). I suppose I am wondering just how a hardcore "factualist", like our midividisage , could ever get to the facts he needs without first having some important beliefs that transcend and rationally order the specific facts themselves. I smell a self-destructive Humean quagmire in the fellow's casting hasty aspersions on belief. After all, it was Hume that unwittingly consigned his own writings to the flames, precisely by concluding his Enquiry Concerning Human Understanding with these words:
Does it [i.e., any book of metaphysics, theology, etc.] contain any abstract reasoning concerning quantity or number? No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Consign it then to the flames: For it can contain nothing but sophistry and illusion.
But does not Hume's own Enquiry begin to burn with these words? His own narrowness about the bounds of metaphysical reasoning backfires and condemns his own reasoning, about the narrowness of metaphysics, as itself a metaphysical construction. This objection is not original with me. Fr. Stanley Jaki has made it in several of his writings and the short article by Scott Lanser, "Commit it to the Flames", makes the same point thus:
…[S]ince his statement is philosophical and not scientific, it deserves to be treated like all the other religious beliefs, writings and statements he condemns: his philosophy too should be cast to the flames!
I think the fellow, in a well-meaning effort to curb the "fanaticism" of Youtube evangelists, is confusing being rational with being logical. The latter task strikes me as much easier to pull off than the former, even though, paradoxically, its scope is much smaller than that of the former. Anyone can spin out logical finger puzzles in their free time; but it takes a lot of work to be rational, a skill that involves logic without being confined to it. I define rationality, at this juncture, at least, as the exclusively human ability to arrange intentions and means toward certain ends, based on a relatively informed knowledge of the contingencies involved in achieving that end; and an important caveat about "relative knowledgeability" for being rational, is that our beliefs are the only things that can illuminate and, in certain ways, determine what we can rationally accept as facts in the first place. Even my belief that I am here as one conscious person is subject to attacks by a radical phenomenalist, like the kind Bertrand Russell channeled from time to time.
Facts, it seems, are really only as handy and indubitable as you believe them to be. Facts, it seems, depend foremost, for an empiricist, on just how skeptical you are willing to be. Everyone likes to think they've found their halcyon 40 acres of reality. The question is, however, primarily that of which worldview, which matrix of beliefs, gives the best grounding for taking data at factual, face-value and working them into logical operations. In my support of the Christian faith as the only really coherent matrix for recognizing, submitting to, and employing factual reality, I am among the ranks of Gilson, Maritain, Jaki, Plantinga, Wolterstoff, Reid, and, among others, St. Thomas. How can one logically construct an empirical world out of mere empirical sensibilia (or, empiricalia )? The Logos is the ground for any logic.
That is all I am able or willing to invest in this quandary at the moment. I believe, however, the facts of the matter will percolate within and generate more logical meanderings somewhere down the road.
 Just as reminder, the minor premise of a modus ponens, such as "Merlin is pleth", would not yield a sound conclusion; it introduces the fallacies of affirming the consequent and an undistributed middle.
 Yeah, I just made that term up. The three lexical segments in it should be clear enough.
 I also just made that term up, I think. Meh. Game on.