These tick values will not have any effect if the y
values themselves are not within the 2-32 range. Right now that chart only has y
values up to 2+.
So, if in addition to adding chart.yAxis.tickValues(['2','4','8','16','32']);
to the Chart Code/Javascript tab as you have in your question, you also make the following change to the code in the Data(JSON) tab...
y: Math.floor(Math.random() * 33) // change from y: random()
...then you will see the tick values you specified (2-32) show up in the y
axis.
NOTE: I assume you are looking at the Scatter / Bubble Chart.