My dishwasher has little racks on the top shelf just for chopsticks.
We also wash our wooden utensils in the dishwasher all the time. They last for years, too.
Cast iron should not go into the dishwasher. I also do not put the wooden salad bowls that I got in Fiji into the dishwasher. Most other stuff goes in, usually without rinsing.
Many dishes and utensils are not recommended for the DW not just because of the prolonged heat, but because dishwasher detergent can be very caustic. Most of it contains some measure of sodium hydrochlorite - bleach. Very destructive on some materials.
I’ve been told that fine silver and porcelain shouldn’t be exposed to dishwasher detergent. The combination of the heat and the caustic cleanser can do a number on it.