Unanswered questions don't necessarily cause cognitive dissonance. Need for closure varies across individuals; some of us don't mind having some (or even many) unanswered questions much at all.
One also moves forward along a path while "looping," and that path isn't necessarily infinite; in fact, it probably isn't for any mortal, practically speaking. For people with relatively low personal need for structure, novelty (and presumably uncertainty) may even be somewhat reassuring when confronting one's mortality (Vess, Routledge, Landau, & Arndt, 2009). Maybe accepting mortality and one's inability to individually pursue the path to completion (regardless of whether completion is possible in any sense) is just another way to not desire the impossible, but it is not impossible to continue the quest for some distance. Depending on what one really wants, not continuing might be a more dissonant alternative to simply continuing as far as one can. If one wants to complete the quest in the sense that one can't, some dissonance may be inevitable; if one wants to participate in the quest without concern for completion, only stopping should cause dissonance with that motive.
Here's an interesting quote from Heflick (2006) relating mortality salience to need for closure:
Landau and colleagues (2004) found that mortality salience increases desire for clear, well structured information and for paintings that have a clear, coherent meaning (Landau et al., 2006). Further, death salience has been found to increase preference for more cultivated landscapes (Koole & Van den Berg, 2003 [published 2005]), for letter sequences that have clear patterns (e.g., AAAABBBB instead of ABBBAABA; Dechesne & Wigboldus, 2002) and stereotypical individuals (Schimel et al., 1998 [1999 in references]). Lastly, these effects are typically stronger among people that are high in epistemic needs (e.g., Landau et al., 2004), including need for closure (e.g., Schimel et al., 2004 [not sure what is being cited here]).
This evidence further suggests (rather indirectly) that some people do find the the thought of uncertainty troubling in light of a threat to their ability to resolve it, but some people don't mind uncertainty all that much, depending partly on individual differences in need for closure. It might even have something to do with your political preferences (Jost, Glaser, Kruglanski, & Sulloway, 2003; Jost et al., 2007)!
Dechesne, M., & Wigboldus, D. (2002). [Terror management in a minimal worldview paradigm.] Unpublished raw data. [As referenced here with year 2001.]
Heflick, N. A. (2006). Terror management and self-enhancement: The moderating role of self-esteem and need for closure (Doctoral dissertation, University of South Florida). Available online, URL: http://scholarcommons.usf.edu/cgi/viewcontent.cgi?article=3005&context=etd. Retrieved January 25, 2014.
Jost, J. T., Glaser, J., Kruglanski, A. W., & Sulloway, F. J. (2003). Political conservatism as motivated social cognition. Psychological Bulletin, 129(3), 339.
Jost, J. T., Napier, J. L., Thorisdottir, H., Gosling, S. D., Palfai, T. P., & Ostafin, B. (2007). Are needs to manage uncertainty and threat associated with political conservatism or ideological extremity? Personality and Social Psychology Bulletin, 33(7), 989-1007. Available online, URL: http://www.researchgate.net/publication/6216707_Are_needs_to_manage_uncertainty_and_threat_associated_with_political_conservatism_or_ideological_extremity/file/9fcfd507eeb6a8980d.pdf. Retrieved January 25, 2014.
Koole, S. L., & Van den Berg, A. E. (2005). Lost in the wilderness: Terror management,
action orientation, and nature evaluation. Journal of Personality and Social Psychology, 88(6), 1014-1028.
Landau, M., Greenberg, J., Solomon, S., Pyszczynski, T. & Martens, A. (2006). Windows
into nothingness: Terror management, meaninglessness and negative reactions to modern art.
Journal of Personality and Social Psychology, 90, 879-892.
Landau, M., Johns, M., Greenberg, J., Pyszczynski, T., Martens, A., Goldenberg, J.L. &
Solomon, S. (2004). A function of form: Terror management and the structuring of the social world. Journal of Personality of Social Psychology, 87, 190-210.
Schimel, J., Simon, L., Greenberg, J., Pyszczynski, T., Solomon, S., Waxmonski, J., &
Arndt, J. (1999). Support for a functional perspective on stereotypes: Evidence
that mortality salience enhances stereotypic thinking and preferences. Journal of
Personality and Social Psychology, 77, 905-926.
Vess, M., Routledge, C., Landau, M. J., & Arndt, J. (2009). The dynamics of death and meaning: the effects of death-relevant cognitions and personal need for structure on perceptions of meaning in life. Journal of Personality and Social Psychology, 97(4), 728. Available online, URL: http://existentialandselfprocesseslab.yolasite.com/resources/Vess_et_al_2009.JPSP.deathmeaning.pdf. Retrieved January 25, 2014.