Summary (copied from last paragraph)
It appears that extended exposure to cognitive dissonance ignites an initial emotional reaction that is then mediated by the rationalization process. If no rationalization can be made, or if the emotion cannot be effectively controlled by the individual, then the initial emotional reaction may grow more powerful, resulting in mood-regulation disorders such as depression, anxiety, or post-traumatic stress disorder. 
Cognitive dissonance is thought to reveal a human's internal desire for consistency. Those who experience cognitive dissonance become psychologically uncomfortable, and thus may take measure to reduce situations that may induce cognitive dissonance. 
Humans have a tendency to want to explain how the world works. There is a discomfort in not 'knowing'. In cognitive dissonance theory, there are several underlying themes, most notably that of religion in cognitive dissonance. Insofar as the theory goes, religion is one helpful mechanism for avoiding cognitive dissonance, as religion often provides an explanation for the nature and generation of the world, as well as a set of guidelines to follow. This provides humans with internal consistency. Skepticism/loyalty to the 'objective' nature of the scientific method can be another form of avoiding cognitive dissonance.  Rejection of any sort of method of navigation (or purpose) in the world often lends itself to a feeling of nihilism, though this is not always related to cognitive dissonance as it is defined.
Sigmund Freud himself observed several instances of what he coined as "Das Unheimliche" -- German for 'the opposite of what is familiar', and translated into English as 'the uncanny'. He describes it as the experience of that which is strangely familiar, not just mysterious. Though the term is not an adequate synonym for 'cognitive dissonance', it is nonetheless relevant in the reaction it produces:
Because the uncanny is familiar, yet incongruous, it has been seen as creating cognitive dissonance within the experiencing subject, due to the paradoxical nature of being simultaneously attracted to yet repulsed by an object. This cognitive dissonance often leads to an outright rejection of the object, as one would rather reject than rationalize, as in the uncanny valley effect.
This initial emotional response is supported by neuroscientific literature when we consider cognitive dissonance as an observable phenomenon. College students who were told to write an argumentative essay in support of a stance that they did not agree with showed increased activity in the anterior cingulate cortex. Next, the left frontal cortex was activated, which is known to activate the 'approach motivational system'.  The motivational directions model asserts that this system regulates one's response to their own anger. If the situation cannot be controlled or changed by the person, they may have less motivation to reduce or regulate their anger, and other emotions may arise. 
Thus, it appears that extended exposure to cognitive dissonance ignites an initial emotional reaction that is then mediated by the rationalization process. If no rationalization can be made, or if the emotion cannot be effectively controlled by the individual, then the initial emotional reaction may grow more powerful, resulting in mood-regulation disorders such as depression, anxiety, or post-traumatic stress disorder. 
 Festinger, L. (1957). A Theory of Cognitive Dissonance. California: Stanford University Press.
 R. A. Wicklund, J. W. Brehm. (1976). Perspectives on Cognitive Dissonance.
 Beauvois, J. L., Joule, R. V., 1996. A radical dissonance theory. London: Taylor and Francis.
 Harmon-Jones, E., Mills, J., Cognitive Dissonance: Perspectives on a Pivotal Theory in Social Psychology. American Psychological Association, Washington, DC, pp. 71–99.
 Hull, A.M. (2002). Neuroimaging findings in post-traumatic stress disorder. The British Journal of Psychiatry 181: 102-10.