As noted by Ulrike Fischer in a comment on Can't apply color inside fancyfoot, setmainfont overwrites it, the issue is that the class sets the color in the main font loading command (\setmainfont), and that takes precedence over any color settings in the document itself. Also noted there is a solution, i.e., define a new font family with a different color and use that in the environment where you want to change the column.
In the code snippet below the original font declaration from Deedy Resume is copied from the class file and used for a new font family with color blue, just to illustrate the effect.
To apply this font family the tightemize is redefined by copying the original definition from the class file and adding \itemfontfamily at the start.
Code:
\documentclass[]{deedy-resume-openfont}
% define font family
\newfontfamily\itemfontfamily[Color=blue, Path = fonts/lato/,BoldItalicFont=Lato-RegIta,BoldFont=Lato-Reg,ItalicFont=Lato-LigIta]{Lato-Lig}
% redefine tightemize environment to use the new font family
\renewenvironment{tightemize}{%
\itemfontfamily%
\vspace{-\topsep}\begin{itemize}\itemsep1pt \parskip0pt \parsep0pt}
{\end{itemize}\vspace{-\topsep}}
\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhf{}
\begin{document}
Result:
