Would a psychopath create a false positive Voight-Kampff test result?

It is unfortunately a very difficult question to answer from the movie alone. Yes, the VK test is challenged in the movie by Rachel - and that it takes very many more questions for Deckard to conclude that she is a Replicant - but that is about the extent of the analysis of the test in the movie. One can imagine from this scenario where a Replicant with artificial human memories almost fools the test, that a human with emotional issues involving empathy may not test as a human. However that (and the opening scene) is the extent of the use of the test in the movie - to introduce empathy as a key difference between humans and replicants, but also to blur the boundary between the two.

The novel from which the movie is loosely adapted (Do Androids Dream of Electric Sheep) the test is discussed in more detail. In the society of the novel most animals in the world have been wiped out, and most if not all humans keep one or more animals as pets. To not care for one or more animals would be unusual. Deckard owns an 'electric sheep' a replica animal, but wishes he could afford a more expensive real pet and is planning to spend his bounty money on one. The fact that he has not to date put enough of a priority on owning a real animal despite his good job is a suggestion in the movie that perhaps Decard's humanity is questioned. As a result the VK test in the novel seems particularly geared towards questions about empathy towards animals, or even insects - as abusing or killing even an insect would be taboo in this society.

In the novel, the issue surrounding the validity of the test, and indeed the validity of Deckard's claim to being human at all - is challenged by his discovery of an entire shadow police department, staffed by what he believes are androids. The blade-runner from this alternative police department uses an entirely different test to distinguish android from human - one that relies on the speed of reflex actions. The two blade runners distrust each other and debate whether one or the other is an android and whether they should take one or other of the tests.

The key thing from both the movie and the novel is the question .. what is human? The validity of the test ceases to be important in both.