Make non-breakable spaces (~) visible in the PDF and preserve their glue

Here is a possible solution:

\documentclass{article}

\usepackage{color}    
\newcommand\testlinei{Lorem~ipsum dolor~sit\linebreak}
\newcommand\testlineii{Lorem~ipsum dolor~sit~skdjfh~skdhf~skfdh~sdf\linebreak}

\begin{document}\noindent\hsize 5cm
\testlinei
\testlineii
\let\oldtilde=~
\def~{\rlap{\textcolor{blue}{$\cdot$}}\oldtilde}
\testlinei
\testlineii
\end{document}

You can use leaders:

\documentclass{article}

\usepackage{color}
\newcommand\testline{Lorem~ipsum dolor~sit\linebreak no underfull}

\renewcommand~{\visiblenobreakspace{}}
\DeclareRobustCommand\visiblenobreakspace{%
  \leavevmode\nobreak
  \cleaders\hbox to 0.3em{\hss\textcolor{blue}{\v{}}\hss}\hskip
  \ifdim\spaceskip>0pt
    \spaceskip
  \else
    \fontdimen2\font plus \fontdimen3\font minus \fontdimen4\font
  \fi
}

\begin{document}

\noindent\hsize 5cm
\testline

\end{document}

With a rule, instead:

\documentclass{article}

\usepackage{color}
\newcommand\testline{Lorem~ipsum dolor~sit\linebreak no underfull}

\renewcommand~{\visiblenobreakspace{}}
\DeclareRobustCommand\visiblenobreakspace{%
  \leavevmode\nobreak
  {\color{blue}%
   \cleaders\hrule height 1ex depth -0.7ex\hskip
   \ifdim\spaceskip>0pt
     \spaceskip
   \else
     \fontdimen2\font plus \fontdimen3\font minus \fontdimen4\font
   \fi
  }%
}

\begin{document}

\noindent\hsize 5cm
\testline

\end{document}

Here are two solutions that center the dot between the two words:

\def\visiblenobreakspacesymbol{$\cdot$}
\let\invisiblenobreakspace=~
\newdimen\fontdimentwo
\newdimen\fontdimenthree
\newdimen\fontdimenfour
\protected\def\visiblenobreakspace{%
    \fontdimentwo=\fontdimen2\font
    \fontdimenthree=\fontdimen3\font
    \fontdimenfour=\fontdimen4\font
    \fontdimen2\font=\dimexpr\fontdimen2\font/2\relax
    \fontdimen3\font=\dimexpr\fontdimen3\font/2\relax
    \fontdimen4\font=\dimexpr\fontdimen4\font/2\relax
    \invisiblenobreakspace
    \hbox to 0pt{\hss\visiblenobreakspacesymbol\hss}%
    \invisiblenobreakspace
    \fontdimen2\font=\fontdimentwo
    \fontdimen3\font=\fontdimenthree
    \fontdimen4\font=\fontdimenfour}
\def~{\visiblenobreakspace}

and

\usepackage{tikz}
\usetikzlibrary{tikzmark}

\newcommand{\visiblenobreakspacesymbol}{$\cdot$}
\let\invisiblenobreakspace=~
\newcounter{nbsp}
\DeclareRobustCommand{\visiblenobreakspace}{%
    \stepcounter{nbsp}%
    \tikzmark{{nbsp-left\arabic{nbsp}}}%
    \invisiblenobreakspace
    \tikzmark{{nbsp-right\arabic{nbsp}}}%
    \tikz[remember picture,overlay]
        \path ({pic cs:nbsp-left\arabic{nbsp}})
            -- ({pic cs:nbsp-right\arabic{nbsp}})
            node [midway,anchor=base] {\visiblenobreakspacesymbol};}
\AtBeginDocument{\renewcommand{~}{\visiblenobreakspace}}

Both produce the following output:

The first solution should run on any TeX engine with e-TeX available. The second solution requires two passes, but it is more powerful; a small change gives the following output:

\usepackage{tikz}
\usetikzlibrary{tikzmark}

\let\invisiblenobreakspace=~
\newcounter{nbsp}
\DeclareRobustCommand{\visiblenobreakspace}{%
    \stepcounter{nbsp}%
    \tikzmark{{nbsp-left\arabic{nbsp}}}%
    \invisiblenobreakspace
    \tikzmark{{nbsp-right\arabic{nbsp}}}%
    \tikz[remember picture,overlay]
        \draw ({pic cs:nbsp-left\arabic{nbsp}}) -- +(0,-0.3ex)
            -| ({pic cs:nbsp-right\arabic{nbsp}});}
\AtBeginDocument{\renewcommand{~}{\visiblenobreakspace}}