Absence of term

It would seem that this is more of an appeal to Biblical literalism rather than theology, as it assumes that the Bible is literally the inerrant and final Christian authority, whereas theological apologetics is permitted a wide berth in interpretation, speculation, connecting the dots, or what Dr. Daniel Dennett describes as theological "spin". 

Dr. Peter Boghossian describes theology as a false epistemology. Dr. Andrew Bernstein defines theology "... in its distilled essence: The employment of high-powered human intellect, of genius, of profoundly rigorous logical deduction—studying nothing.”

 So, the claim in the OP's question: "The doctrine of the Trinity is false as it is never mentioned in the Bible" is precisely what you'd expect from biblical literalism and not theology. The doctrine of the Trinity is exactly what you would expect from theological musings and sophistry.

So, the fallacy is a bit of equivocation as it mistakenly inserts the term theology for literalism and concludes with a fallacious idea that any alternative to literal interpretation is false.

Hi James,

The pattern itself you suggest that might be a fallacy is very specific to certain contexts, which is why it wouldn't be a fallacy. The potential problem with the reasoning here (in your given example) is the overall assumption that if some Christian doctrine isn't specifically and clearly discussed in the Bible, that it is not true. This isn't a question of logic nor reason, but one of faith or theology. One might, through faith, believe that any religious concept that is true would be clearly stated in the Bible, and through their specific interpretation, they would deny that the trinity is clearly stated, therefore, false.

I don't meant to sound preachy, but when dealing with magic, gods, myths, supernatural, "the Holy Spirit making it clear" to someone, or anything not subject to reason and logic, anything goes.

You might be able to introduce reason by holding the person accountable for being at least internally consistent . For example, ask them if they believe that any religious concept that is true would be clearly stated in the Bible. Then find a religious belief they hold that is not clearly stated in the Bible.