'The fallacy fallacy' is making the assumption that the conclusion is wrong because the argument is wrong. A wrong argument can still have a correct conclusion.
1. The Eggs of the Duck Billed Platypus
All fish lay eggs.
The duck-billed platypus is a fish.
Therefore the duck-billed platypus lays eggs.
This is a valid argument with two false premises and a true conclusion.
Premise one is false because some fish give birth to live young.
Premise two because the duck-billed platypus is certainly not a fish.
The conclusion, however, is true since duck-billed platypuses do lay eggs.*
2. The Art Gallery Fee
Some art galleries don’t charge an entrance fee.
London’s National Gallery is an art gallery.
Therefore London’s National Gallery doesn’t charge an entrance fee.
The premises of this argument are true; and it is true that London's National Gallery doesn’t charge an entry fee.
Yet this conclusion does not reliably follow from the premises since they leave open the possibility that London's National Gallery might charge an entry fee.
All that the first premise tells us is that some art galleries are free; it gives no clues as to whether or not London's National Gallery falls within the class of free galleries.
This is a weakness in the way the conclusion has been reached.*
3. The Dodgy Study and the Computer Illiterate
A poorly conducted piece of sociological research designed to assess the causes of criminal behaviour might, despite being based on an unrepresentative sample and inappropriate statistical tests, turn up some true conclusions.
Someone who knows next to nothing about computers might correctly identify that your disk drive is faulty even though the way they arrived at this conclusion involved all kinds of reasoning errors.
Poor reasoning in no way guarantees false conclusions. So, in order to refute a conclusion it is not enough simply to show that it has been reached by unreliable means; you need to provide further argument that demonstrates that it is false.*
4. The Dinosaur Rider
Karen: I am sorry, but if you think man used to ride dinosaurs, then you are obviously not very well educated.
Kent: First of all, I hold a PhD in creation science, so I am well-educated. Second of all, your ad hominem attack shows that you are wrong, and man did used to ride dinosaurs.
Karen: Getting your PhD in a couple months, from a “college” in a trailer park, is not being well-educated. My fallacy in no way is evidence for man riding on dinosaurs, and despite what you may think, the Flintstone’s was not a documentary!
Karen’s ad hominem fallacy in her initial statement has nothing to do with the truth value of the argument that man used to ride dinosaurs.**
5. The Methodological Flaw
This new test seemed so promising, but the three studies that supported its validity turned out to have critical methodological flaws, so the test is probably not valid. 
At times, fallacies are used by those who can’t find a better way to support the truth claims of their argument—it could be a sign of desperation. This can be evidence for them not being able to defend their claim, but not against the claim itself. 
* * *
*Thinking from A to Z by Nigel Warburton
**Logically Fallacious: The Ultimate Collection of Over 300 Logical Fallacies by Bo Bennett
 Ethics in Psychotherapy and Counseling: A Practical Guide by Kenneth S. Pope, Melba J. T. Vasquez
Note: It's also called: “argumentum ad logicam” “argument from hearsay”, “bad reasons fallacy”, “argument from fallacy”, “argument to logic”.
Categorisation: Philosophy → Logic → logical fallacy → formal
* * *