Since brain chemistry is an incredibly complex, vast, and, most importantly, unknown topic, nobody can answer the question itself. Brain chemicals show interdependencies much beyond our current understanding, especially with regards to psychological states and conditions. So we cannot answer the question much beyond saying: maybe. But what we can do is evaluate the specific claims the author (Feinstein) makes, and see what the references he is providing for them state.
Feinstein provides two references to studies he interprets as discussing the beneficial effect of "acupoints" on brain chemistry.
The second, Church, D., Yount, G., & Brooks, A. (in press). The effect of Emotional Freedom Techniques (EFT) on stress biochemistry: A randomized controlled trial. Journal of Nervous and Mental Disease., is reported to show that "the treatment significantly reduced cortisol levels". This is indeed what Church et al. report; cortisol levels fell on average by ~25%. It fell by only ~15% in the alternative treatment group (a therapeutic interview). However, it also fell by 15% in the control (no treatment) group, so basically, they compared "EFT" (supposedly similar to "acupoints") to something that had no meaningful effect on cortisol. However, in itself, touching leading to cortisol decrease is not surprising, as e.g. massages lead to lowered cortisol levels - by, it seems, on average over 30%.
Furthermore, cortisol is not a prototypical "brain chemical". Its main effect is probably regulating blood sugar, and generally nutrient metabolism. It can be used as an index of stress, but a slight and short-term decrease in cortisol is not necessarily what I'd universally consider a "desirable change in brain chemistry". However, it can be a used as a proxy of transient relaxation.
The first reference is much more vague; Feinstein writes:
Biochemical effects of acupoint stimulation are also being identified, with neurotransmitters, endorphins, and other brain chemicals apparently being influenced by tapping (Ruden, 2005).
Looking up Ruden, we find:
we would like to speculate about a potential mechanism for tapping of the fear response
And that is indeed what he does. Ruden speculates, which is okay, but that's the extent of it. He cites no direct research on the question itself, reports no actual empirical results of direct tests of his ideas. So, "do acupoints decrease desirable changes in brain chemistry"? Maybe.
Finally, let it be said that the Feinstein paper presents some truly horrible statistical methods and is an insult to the idea of meta-analysis.