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.