In my opinion, the cleanest way to remove all tick labels for the y axis is to use yticklabel=\empty or yticklabel={\empty} (xticklabel=\empty would do the same for the x axis). Note that there is no s in yticklabel!
This works because the /pgfplots/xticklabel, /pgfplots/yticklabel and /pgfplots/zticklabel keys accept a {〈code〉} argument that defines how to format each tick in a generic way—see the excerpt from the pgfplots manual reproduced below.
\documentclass{article}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\begin{document}
\begin{tikzpicture}
\begin{axis}[
title={Test axis},
xlabel={Test $x$ label},
ylabel={Test $y$ label},
yticklabel=\empty,
]
\end{axis}
\end{tikzpicture}
\end{document}

Quote from the pgfplots manual
What follows is the beginning of the documentation for the /pgfplots/xticklabel, /pgfplots/yticklabel and /pgfplots/zticklabel keys in the pgfplots manual, version 1.16 (latest release as of January 2020).
/pgfplots/xticklabel={〈command〉}
/pgfplots/yticklabel={〈command〉}
/pgfplots/zticklabel={〈command〉}
These keys change the TeX command which creates the tick labels assigned to each tick position (see options xtick and ytick).
This is one of the two options to assign tick labels directly. The other option is xticklabels={〈label list〉} (or yticklabels={〈label list〉}). The option xticklabel offers higher flexibility while xticklabels is easier to use.
The argument 〈command〉 can be any TeX text. The following commands are valid inside of 〈command〉:
\tick The current element of option xtick (or ytick).
\ticknum The current tick number, starting with 0 (it is a macro containing a number).
\nexttick This command is only valid if the x tick label as interval option is set (or the corresponding variable for y). It will contain the position of the next tick, that means the right boundary of the tick interval.
The default argument is
\axisdefaultticklabel for normal plots:
\def\axisdefaultticklabel{$\pgfmathprintnumber{\tick}$}
\axisdefaultticklabellog for log plots:
\def\axisdefaultticklabellog{%
\pgfkeysgetvalue{/pgfplots/log number format code/.@cmd}\pgfplots@log@label@style
\expandafter\pgfplots@log@label@style\tick\pgfeov
}
But unfortunately it didn't help. Any other ideas?
– eliasp Nov 28 '10 at 20:10