Remove space before itemize

Remove the faulty \\ after #1 -- it does no good there. Leave an empty blank line to separate the items. In my point of view this however no really convincing way to use itemize etc.

\documentclass{article}
\usepackage{enumitem}
\setlength{\parindent}{0pt}

\newcommand{\NewEntry}[2]{%
    #1% 

    #2%
}

\begin{document}
    \NewEntry{ENTRY}
    {\begin{itemize}[topsep=0pt]
    \item First Item
    \item Second Item
    \end{itemize}
    }
    \NewEntry{ENTRY}{%
      \begin{itemize}[topsep=0pt]
    \item First Item
    \item Second Item
    \end{itemize}
    }
\end{document}

Some suggestions to improve this

\documentclass{article}
\usepackage{xparse}
\usepackage{enumitem}
\setlength{\parindent}{0pt}

\newlist{notopsepitemize}{itemize}{1}
\setlist[notopsepitemize,1]{label={\textbullet},topsep=0pt}


\ExplSyntaxOn
\clist_new:N \l_tonychief_item_clist
\NewDocumentCommand{\NewEntry}{O{}mm}{
\clist_set:Nn \l_tonychief_item_clist {#3}
    #2

    \begin{notopsepitemize}[#1]
    \item 
      \clist_use:Nn \l_tonychief_item_clist {\item} %
    \end{notopsepitemize}
    }
\ExplSyntaxOff

\begin{document}
    \NewEntry{ENTRY}{Item One, Item Two, Item Three, Item Four}%

    \NewEntry[topsep=20pt]{Other ENTRY}{Item One, Item Two, Item Three, Item Four}%
\end{document}

If it is to be used with itemize like this, here is an enumitem way using before=Entry

\documentclass{article}
\usepackage{enumitem}
\setlength{\parindent}{0pt}

\begin{document}
    \begin{itemize}[topsep=0pt,before=Entry]
    \item First Item
    \item Second Item
    \end{itemize}
\end{document}

I agree with @Christian that a \par is better than \\ and that in general it seems a rather odd way of constructing your \NewEntry command.

However just for reference, you may use a negative space in the argument with the itemize:

\NewEntry{ENTRY}
{\vspace{-\baselineskip}
\begin{itemize}[topsep=0pt]
\item First Item
\item Second Item
\end{itemize}
}

Still, I would not recommend such a construction.

Try instead to see if you could make your command more "semantic".