Soldering iron to make holes in sticks?
Plastic will melt and flow away from the iron leaving a hole.  Cardboard is really just paper with air gaps between the glued layers so there's just a little bit of material to burn.

A wooden stick is going to be solid wood.  It doesn't melt and move out of the way - if anything the burned area of wood actually gets harder (and more brittle) as it turns to charcoal.  You're not going to just burn a hole through it.

The right tool is, of course, a drill.  You may be able to use a sharply pointed knife for a smaller stick, but a drill really is going to be your best bet.  It will also work for the plastic and cardboard, and in my opinion, will be safer than using a burning hot iron.