How do I prevent conflicts between accsupp and hyperref?

I would avoid to redefine \thelstnumber in such a way. The command can be used in various places where such fragile content can break easily. In the case of listings I would suggest to use the key numberstyle instead:

\documentclass{article}

\usepackage{listings}
\usepackage{accsupp}

\newcommand\emptyaccsupp[1]{\BeginAccSupp{ActualText={}}#1\EndAccSupp{}}

\lstset
{
    language={[LaTeX]TeX},
    numbers=left,
    columns=flexible,
    numberstyle=\emptyaccsupp
}

\usepackage{hyperref}

\begin{document}

\begin{lstlisting}
Hello\\
Universe!
\end{lstlisting}

\end{document}

It seems hyperref expands the \thelstnumber in an context which does not set \protect correctly (e.g. a plain \edef) which breaks the fragile code. You can fix this by defining a \theHlstnumber which is used by hyperref. This macro is apparently set at the begin of the document by hyperref and you so you need to redefine it afterwards. Note that the original definition includes also a leading \ifx \lst@@caption \@empty \lst@neglisting \else \thelstlisting \fi, which you might want to add as well.

\documentclass{article}

\usepackage{listings}
\usepackage{accsupp}
\renewcommand*{\thelstnumber}{\protect\BeginAccSupp{ActualText={}}\arabic{lstnumber}\protect\EndAccSupp{}}

\lstset
{
    language={[LaTeX]TeX},
    numbers=left,
    columns=flexible,
}

\usepackage{hyperref}
\AtBeginDocument{\renewcommand*{\theHlstnumber}{\arabic{lstnumber}}}% or manually put it after `\begin{document}`
\begin{document}
\begin{lstlisting}
Hello\\
Universe!
\end{lstlisting}
\end{document}