How to deal with bad line-wrapping of \texttt?

Depending on how you want the variables to be broken, you can define a macro that allows breaking with certain rules. Here I just allowed breaking after _. Apart from that, you use a more semantic command that you can redefine with flexibility.

\documentclass{report}

\newcommand*\ttvar[1]{\texttt{\expandafter\dottvar\detokenize{#1}\relax}}
\newcommand*\dottvar[1]{\ifx\relax#1\else
  \expandafter\ifx\string_#1\string_\allowbreak\else#1\fi
  \expandafter\dottvar\fi}

\begin{document}

Most of the time, \LaTeX{} line-wrapping works fine. However, sometimes when
I'm writing about code with lots of \ttvar{long_monospace_variables},
\ttvar{the_line_wrapping} doesn't quite get the position correct resulting in
slightly-too-long lines.

\end{document}

Based on the answers to this question

Line wrapping on narrow pages

it seems that the best way to do this is to wrap the offending paragraph in a sloppypar environment.

\documentclass{report}
\begin{document}
\begin{sloppypar}
Most of the time, LaTeX line-wrapping works fine.  However, sometimes when I'm writing about code with lots of \texttt{long\_monospace\_variables}, \texttt{the\_line\_wrapping} doesn't quite get the position correct resulting in slightly-too-long lines.
\end{sloppypar}
\end{document}

Using a \sloppy command has a similar effect for the whole document, but this seems a little dangerous since it could break formatting elsewhere.

It's easy to do with the url package. I define a \longvar command which does the job:

\documentclass{report}

\usepackage{url}

\newcommand\longvar[1]{\mathchardef\UrlBreakPenalty=100
\mathchardef\UrlBigBreakPenalty=100\url{#1}}

 \begin{document}

Most of the time, LaTeX line-wrapping works fine. However, sometimes when I'm writing about code with lots and lots of \longvar{very_long_monospace_variables}, \texttt{the\_line\_wrapping} doesn't quite get the position correct resulting in slightly-too-long lines.

\end{document}