I often run into a phenomenon I've come to call "paradigm-dependent argument" -- there may be a philosophically proper name for it out there somewhere, but I haven't yet heard it.
There are two variants -- paradigm dependent evidence for a theory, and paradigm-dependent refutations of a theory. Both are related to, but not identical to, the informal fallacy of "Begging the question"
Paradigm-dependent evidence works like this. In order to provide evidence for a theory, you line up evidence; but the significance of the evidence in supporting the theory depends on assuming the theory is true in the first place.
For example: "The Bible is inerrant. As evidence, there is a passage in which it claims to be God-breathed, and we know God cannot breath lies!"
The evidence to support the conclusion of inerrancy is the passage in which "all scripture" is described as "God-breathed." However, the significance of this "evidence" depends on the assumption that the Bible is inerrant. If the Bible is not inerrant, then this passage could well be in error. And if the passage is in error, then it holds no significance in supporting the conclusion.
On the other side of the ideological chasm, evolutionists often use paradigm-dependent evidence to support the conclusion of common descent. For example, "Human and chimp DNA is overwhelmingly similar; therefore humans and chimps are closely related." However, this "evidence" depends for its significance on the assumption that humans and chimps are related. If humans and chimps are not related, then it's entirely possible that the designer or designers simply used very similar designs for the two, modifying the designs only insofar as necessary to achieve the desired differences.
The end result is that people can appear to be lining up all sorts of "evidence" to support a conclusion, but because of the nature of the evidence itself, it's not really evidence at all.
This is basically the other side of the coin. I present evidence against your theory which depends for its significance upon the assumptions of my theory, or a strawman version of your theory.
The one I run into most often is the argument against design from "suboptimal design." It goes like this: Many aspects of the biological design of humans are less efficient than they could be. A competent designer would not design things so inefficiently. Therefore they were not designed.
This "evidence" depends on a straw-man assumption about design -- that the designer, must, of necessity, have intended to create life with optimal efficiency. But without that assumption, we're left with the possibility that life was designed to be efficient, but not perfectly efficient. Much like engineers today, compromises in efficiency have to be made to get the dang thing to work.
Therefore, the evidence depends for its significane on the tacit assumptions that "The Designer would have designed optimally," and "our conclusions of suboptimal designer are actually suboptimal." Without these two dubious assumptions, the "evidence" holds no significance.
The funniest thing of all is hearing scientists call the "design of life" suboptimal, when it is so vastly beyond our ability to design, construct, or even understand. Because we can imagine life being "better designed," (even though we can't even design or construct the most basic forms), we infer that the "Designer" was incompetent. Talk about Monday-morning, arm-chair quarterbacking.
A theistic example of this fallacy is the ontological argument. It goes like this: God is defined as "The being than which no greater can be conceived." It is greater to exist than not to exist. Therefore, if you conceive of God as not existing, you're conceiving of something less than "the greatest." Therefore, if you conceive of God as not existing, you're not really conceiving of God. Therefore, atheism is self-contradictory.
This argument depends on the assumption that "God is defined as 'the being than which no greater can be conceived.'"
But if an atheist instead conceives of God as "An imaginery conception in the mind of men without any corresponding reality outside the mind," then the ontological argument holds no power. The whole argument depends on a definition of God with which an atheist could reasonably disagree.