In very simple words, Argument from Respect for Authority or Argumentum Ad Verecundiam is about mentioning an authoritative figure to get something done.
Thomas Cathcart & Daniel Klein shared few good jokes to explain this fallacy in their book, Plato and a Platypus Walk into a Bar. They say that the argument from respect for authority is one of our boss’s favorite arguments. Citing an authority to support your argument is not a logical fallacy in and of itself; expert opinion is legitimate evidence alongside other evidence. 
Testers are obviously not spared of this fallacy. Whenever there is an opportunity, we cite someone from the senior management to support our argument. It is not uncommon to hear, “The best practice of testing must be followed by all projects for all test phases as this is the mandate by the Director of Silly Processes”. Once the director’s mandate is cited, people normally follow the direction no matter whether the director actually said that or not. It also does not matter whether that testing process was actually useful in that context. I am sure most would not go and confirm with the director. After all, who wants to get into trouble with one anyway!
The problem occurs when this argument is used to either cover the behind or to take undue advantage of a system, person or process. I can think of multiple scenarios in this case:
1. The authority cited may not be an expert of the subject matter and the outcome after citation may prove wrong.
2. The authority cited may get challenged by others who are at the same level as her and prove her wrong.
3. Office politics and/or inter-departmental politics may make the whole situation worse affecting the employee morale.
4. Some disgruntled employee may challenge the authority for the accuracy of her assumption. Again, if proven correct, this will make the leadership look weaker. 
5. Probably the worst one is that when the authority cited is not even aware that she has been cited. 
We can add many more scenarios to this list. But as testers what is important for us is the awareness that this is not always in our best interest to give an argument from respect for authority. I prefer either keeping quite or buying time to find an answer if I am not aware of something. Unnecessary arguments such as this one can only diminish one’s reputation even if in the short term it feels like a gain.