This seems very similar to the experiment where black people were shown to do worse on intelligence tests after being reminded that they were black..

So this experiment (in my view) doesn't really help to answer whether analytical thinking reduces religious belief. What it does show is that a lot of people make that association, and that is more than enough to cause the priming effect.

"Cognitive style and religiosity: The role of conflict detection", Pennycook et al 2014; abstract:

Recent research has indicated a negative relation between the propensity for analytic reasoning and religious beliefs and practices. Here, we propose conflict detection as a mechanism underlying this relation, on the basis of the hypothesis that more-analytic people are less religious, in part, because they are more sensitive to conflicts between immaterial religious beliefs and beliefs about the material world. To examine cognitive conflict sensitivity, we presented problems containing stereotypes that conflicted with base-rate probabilities in a task with no religious content. In three studies, we found evidence that religiosity is negatively related to conflict detection during reasoning. Independent measures of analytic cognitive style also positively predicted conflict detection. The present findings provide evidence for a mechanism potentially contributing to the negative association between analytic thinking and religiosity, and more generally, they illustrate the insights to be gained from integrating individual-difference factor

...Empirical studies based on such theorizing have reported that individual differences in cognitive style predict reasoning performance over and above cognitive ability (Stanovich & West, 2000). Moreover, in a noteworthy example of independent convergence, three recent studies have shown that cognitive style has pervasive links not only with reasoning performance on highly constrained laboratory tasks, but with entire belief systems (Gervais & Norenzayan, 2012; Pennycook, Cheyne, Seli, Koehler, & Fugelsang, 2012; Shenhav, Rand, & Greene, 2012). Specifically, individuals who are more likely to engage in analytic processing during problem solving are less likely to hold specific religious and paranormal beliefs, regardless of cognitive ability and numerous demographic and personality variables (see also Cheyne & Pennycook, 2013; Pennycook, Cheyne, Koehler, & Fugelsang, 2013). Increasing analytic processing via experimental manipulation has also been shown to decrease religious belief (Gervais & Norenzayan, 2012; Shenhav et al., 2012).

...Pennycook et al., (2012) hypothesized that Type 2 processing negatively affects religious beliefs because such beliefs are vulnerable to conflicts between the immaterial and material worlds (i.e., based on folk mechanics, folk biology, and folk psychology; Atran & Norenzayan, 2004). Atran and Norenzayan, for example, stated that “conceptions of the supernatural invariably involve the interruption or violation of universal cognitive principles that govern ordinary human perception and understanding of the everyday world” (p. 714). Thus, belief in beings that can pass through solid objects (such as angels) is a violation of folk mechanics (i.e., the belief that objects cannot pass through solid objects; see Boyer, 1994), and so on. It follows from this perspective that more-analytic people may be less religious, in part, because they are more efficient at detecting, or more responsive to, such conflicts. Implicit in this hypothesis is that religious believers will be less likely to detect conflict between two cognitive outputs even in a reasoning task that does not involve religious belief.

Cute results make me wary, but it apparently had 650 participants. Will be interesting to see the effect size.

Article is now out.