Evidence on this question stems from research on ego depletion and the "Strength Model of Self Control" (Baumeister, Vohs, & Tice, 2007). The theory posits that successful self-control relies on a limited resource that is depleted by tasks requiring self control.
In series of experiments, Vohs and colleagues (2008) have investigated the hypothesis that, because making choices is mentally taxing, subsequent attempts to engage in self-regulation are more likely to fail.
Citing from their abstract:
The current research tested the hypothesis that making many choices impairs subsequent self-control. Drawing from a limited-resource model of self-regulation and executive function, the authors hypothesized that decision making depletes the same resource used for self-control and active responding. In 4 laboratory studies, some participants made choices among consumer goods or college course options, whereas others thought about the same options without making choices. Making choices led to reduced self-control (i.e., less physical stamina, reduced persistence in the face of failure, more procrastination, and less quality and quantity of arithmetic calculations). A field study then found that reduced self-control was predicted by shoppers’ self-reported degree of previous active decision making. Further studies suggested that choosing is more depleting than merely deliberating and forming preferences about options and more depleting than implementing choices made by someone else and that anticipating the choice task as enjoyable can reduce the depleting effect for the first choices but not for many choices.
So judging from this research, if you stand in front of your wardrobe every morning and experience the decision between the black and the grey t-shirt as an arduous and mentally taxing choice exercise, it might indeed be better to always wear grey, as Mark Zuckerberg does.
It has to be said, however, that some people have doubted whether self-control can really be understood as a finite resource that can be depleted and favor a motivational account of the effects (e.g., Inzlicht, Schmeichel, & Macrae, 2014; Job, Dweck, & Walton, 2010).
Furthermore, the specific data about making decisions is only the result of one paper (I am not aware of independent replications, but they might exist) and, more generally, there are some doubts about how strong and reliable ego depletion effects are (see this earlier question).
In conclusion, it doesn't hurt to avoid complicated but trivial choices. Nevertheless, I would still hold on to those black shirts (my heuristic is to take the one that lies on top, by the way).
Baumeister, R. F., Vohs, K. D., & Tice, D. M. (2007). The strength model of self-control. Current Directions in Psychological Science, 16(6), 351–355. http://doi.org/10.1111/j.1467-8721.2007.00534.x
Inzlicht, M., Schmeichel, B. J., & Macrae, C. N. (2014). Why self-control seems (but may not be) limited. Trends in Cognitive Sciences, 18(3), 127–133. http://doi.org/10.1016/j.tics.2013.12.009
Job, V., Dweck, C. S., & Walton, G. M. (2010). Ego depletion--Is it all in your head?: implicit theories about willpower affect self-regulation. Psychological Science, 21(11), 1686–1693. http://doi.org/10.1177/0956797610384745
Vohs, K. D., Baumeister, R. F., Schmeichel, B. J., Twenge, J. M., Nelson, N. M., & Tice, D. M. (2008). Making choices impairs subsequent self-control: A limited-resource account of decision making, self-regulation, and active initiative. Journal of Personality and Social Psychology, 94(5), 883–898. http://doi.org/10.1037/0022-35184.108.40.2063