Because biologists disagree about the details of phylogenies, some nonscientists argue that such disagreements warrant the questioning of evolutionary theory in its entirety. Why is this reasoning flawed?

Answer:

This reasoning is flawed because it does not make sense that disagreements within a particular topic should make the entire body of work false. If every topic/subject that had some sort of disagreement between professionals was considered false then basically every single topic of study should then be considered false. For example, in pharmacology there are disagreements on drug/chemical properties all the time, however this does not mean that the drug or chemical does not exist or will not work for a certain purpose. Any subject you name will have disagreements but that does not mean the entire subject is invalid.
To put it in simple terms: just because people don't agree on things doesn't make everything a lie.
The evolutionary theory has many other components, not just phylogenies, so it doesn't make any sense that conflicting ideas means the entire evolutionary theory is a lie. It is a very VERY broad assumption to make, and in my opinion, a very dangerous and narrowminded one (that bit is a personal opinion though). There is plenty of other evidence that supports the evolutionary theory (carbon dating, ice cores, DNA/genetic evidence etc.).