This is a work around using \clip
.
\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
\clip (1.3,0.2) -- (1,0) -- (1.3,-0.2) -- (1.3,0.2) -- cycle;
\draw [line width=2pt] (1.3,0.2) to (1,0) to (1.3,-0.2);
\draw (1.3,-0.2) to (1.3,0.2);
\end{tikzpicture}
\end{document}

Note that a line width of 2pt
will give a resultant line width of 1pt
as the other part is clipped off. So you have to adjust properly.
If this forms a part of bigger diagram, the \clip
ping part may be enclosed in a scope
so as to limit the clipping:
\begin{scope}
\clip (1.3,0.2) -- (1,0) -- (1.3,-0.2) -- (1.3,0.2) -- cycle;
\draw [line width=2pt] (1.3,0.2) to (1,0) to (1.3,-0.2);
\draw (1.3,-0.2) to (1.3,0.2);
\end{scope}
You can specify the line width
. Here is a zoomed in view of the output so that one can see the difference:

Notes:
As John Kormylo pointed out the tikz-pgf
manual says about the "0 width" line:
Line width: The “thickness” of the line. A width of 0 is the thinnest width renderable on the device. On a high-resolution printer this may become invisible and should be avoided. A good choice is 0.4pt, which is the default.
Obviously at some point the size difference won't be noticeable at all, especially in a print edition.
Code:
\documentclass{article}
\usepackage{siunitx}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[yscale=0.5]
\draw [line width=0.25mm, red ] (0,-1) -- (2,-1) node [right] {\SI{0.25}{\milli\meter}};;
\draw [line width=0.1mm, blue] (0,-2) -- (2,-2) node [right] {\SI{0.10}{\milli\meter}};;
\draw [line width=0.05mm, red ] (0,-3) -- (2,-3) node [right] {\SI{0.05}{\milli\meter}};
\draw [line width=0.01mm, blue] (0,-4) -- (2,-4) node [right] {\SI{0.01}{\milli\meter}};
\draw [line width=0mm, black] (0,-5) -- (2,-5) node [right] {\SI{0.0}{\milli\meter}};
\end{tikzpicture}
\end{document}
Best Answer
This is possible but it's not easy and the control of the line width is not very fine but it's an idea and I think it's possible to get a better code. The code below is from an idea of Mark Wibrow to change the color of a line. I modified the code to change the width but if you only want to change the color this code is perfect :
To modifiy this code you need to adapt
\pgfsetlinewidth{\x/40*0.075pt+\pgflinewidth}