It's apparently a problem with the version of Times New Roman you have on your machine, because with mine the “apostrophe” appears. The dashed rectangle is a sign that a character is missing from the font.
If you want to remove it from the particular environment, you can recycle an answer of mine; it was written for polyglossia and xgreek, but works also here.
\documentclass[12pt]{report}
\usepackage{fontspec}
\usepackage[english,greek]{babel}
\usepackage[inline]{enumitem}
\usepackage{etoolbox}
\setmainfont{Times New Roman}
% recycled from https://tex.stackexchange.com/a/292127/4427
\makeatletter
\renewrobustcmd{\anw@true}{\let\ifanw@\iftrue}
\renewrobustcmd{\anw@false}{\let\ifanw@\iffalse}\anw@false
\newrobustcmd{\noanw@true}{\let\ifnoanw@\iftrue}
\newrobustcmd{\noanw@false}{\let\ifnoanw@\iffalse}\noanw@false
\renewrobustcmd{\anw@print}{\ifanw@\ifnoanw@\else\numer@lsign\fi\fi}
\newrobustcmd{\noanw}{\noanw@true}
\makeatother
\begin{document}
\begin{enumerate}[label = \noanw\alph*)]
\item first line
\item second line
\end{enumerate}
\end{document}

Note that \selectlanguage{greek} is not needed if you load greek last.
Also, enumerate and enumitem fight each other, so just one of them should be loaded (the latter is much more powerful).
Final recommendation: fontspec should be loaded before babel when Greek is involved.

Alternative solution if you want to remove the “apostrophe” altogether from your document
\documentclass[12pt]{report}
\usepackage{fontspec}
\usepackage[english,greek]{babel}
\usepackage[inline]{enumitem}
\usepackage{etoolbox}
\setmainfont{Times New Roman}
% see https://tex.stackexchange.com/a/173572/4427
\renewcommand\textdexiakeraia{}
\begin{document}
\begin{enumerate}[label = \alph*)]
\item first line
\item second line
\end{enumerate}
\end{document}
\documentclassup to\end{document}. – egreg Oct 09 '18 at 22:14\usepackage{enumerate}(you shouldn't load both it andenumitem), but it's not relevant. Do you get some message in the log file aboutMissing character? – egreg Oct 09 '18 at 22:25