Witkowski has written several articles reviewing NLP (2010, 2012) and in each case has found few studies that indicate clear support for the technique. He also argues that the articles which indicate no support for NLP are generally of stronger methodological quality than those which do not. The fast phobia cure is not directly mentioned in either paper.
Jemmer (2005) provides an overview of the fast phobia technique, and describes a handful of studies which have used and reported on it; you may find his paper to be a starting point for identifying information on the technique, although it seems most of the studies he cited are in the NLP database and it is unclear if they have been published in peer review journals. Jemmer's article acknowledges the lack of evidence, and is primarily written from a theoretical perspective.
A 2011 review identified three peer-reviewed studies and 1 dissertation that examined NLP techniques for phobia and reported the techniques to be effective (Biswal & Prusty). The review does not go into detail as to the exact NLP techniques used in these studies but this may be a source of literature for further reading. Biswal and Prusty (2011) also highlight some of the theoretical issues which have hampered NLP research and provide a helpful overview distinguishing some of NLP's techniques from other psychotherapeutic techniques.
A 2012 review examined 10 peer-reviewed NLP studies which were either randomized controlled trials or pre-post designs, 3 of which assessed fast phobia techniques (Sturt et al.). Some effects were found in terms of decreased fear or anxiety, but these generally occurred in all treatment groups (not just NLP) or in studies with no control group. Most outcomes were self-report measures. However, these studies may be the most relevant to your initial question.
Overall, the consensus of these reviews indicates that there is little evidence to support the use of NLP in general, including the fast phobia cure. However, the studies also note that the lack of high quality research may indicate more rigorous studies are needed before discarding the NLP concept entirely. Additionally, the reviews point out that several NLP phobia techniques are similar to other psychotherapy techniques for treating phobia, so there may be stronger evidence for those subcomponents elsewhere in the literature. These reviews should provide a good starting point and working vocabulary if you are interested in researching those subcomponents.
Biswal, R., & Prusty, B. (2011). Trends in neuro-linguistic programming (NLP): A critical review. Social Science International, 27(1), 41-56.
Jemmer, P. (2005). Phobia: Fear and loathing in mental spaces. European Journal of Clinical Hyponosis, 6(3), 24-32.
Sturt, J., Ali, S., Robertson, W., Metcalfe, D., Grove, A., Bourne, C., & Bridle, C. (2012). Neurolinguistic programming: A systematic review of the effects on health outcomes. British Journal of General Practice, November, e757-e764.
Witkowski, T. (2010). Thirty-five years of research on neuro-linguistic programming. NLP Research Database. State of the art or pseudoscientific decoration? Polish Psychological Bulletin, 41(2), 58-66.
Witkowski, T. (2012). A review of research findings on neuro-linguistic programming. The Scientific Review of Mental Health Practice, 9(1), 29-40.