Hide tick numbers in a TikZ/PGF axis environment.

Just add yticklabels={,,} to the options for the axis environment. Your example then looks like shown below.

\documentclass{scrartcl}
\usepackage{tikz}
\usetikzlibrary{plotmarks}
\usepackage{pgfplots}
\begin{document}
  \begin{tikzpicture}
    \begin{axis}[%
      title={Test Axis},
      xlabel={Test X Label},
      ylabel={Test Y Label},
      yticklabels={,,}
    ]
    \end{axis}
  \end{tikzpicture}
\end{document}

This feature is documented in Section 4.14.2 (Tick Alignment: Positions and Shifts) on page 180 of the pgfplots 1.4.1 manual. Quite hidden in the last example of the mentioned section.

The line

\pgfplotsset{ticks=none}

removes both ticks an labels.

Regarding hiding only the tick labels: one idea is to set the tick label color to the backround color. For example:

\pgfplotsset{tick label style={color=white},
  label style={font=\small},
  legend style={font=\small}

To shift the axis label, additionally modify the corresponding style, for example:

\pgfplotsset{every axis x label/.style={
  at={(0.5,0)},
  below,
  yshift=-5pt}}

Adjust the yshift parameter. Similar for the y axis.

\pgfplotsset{every axis y label/.style={
  at={(0,0.5)},
  xshift=-15pt,
  rotate=90}}

I've stumbled upon a nice solution using \empty as argument for the labels you want to hide, e.g. yticklabels=\empty.

\documentclass{scrartcl}
\usepackage{tikz}
\usetikzlibrary{plotmarks}
\usepackage{pgfplots}
\begin{document}
  \begin{tikzpicture}
    \begin{axis}[
      title={Test Axis},
      xlabel={Test X Label},
      ylabel={Test Y Label},
      yticklabels=\empty,
    ]
    \end{axis}
  \end{tikzpicture}
\end{document}

The above produces