You can put anything you want to into a dishwasher, as long as it will fit. 
That said, I would advise against washing anything made of wood (knife handles, cutting boards, wooden utensils, etc.) in a dishwasher, for all of the reasons given above.
This is experience talking, BTW. My mother regularly put all of her flatware and utensils in the dishwasher so that she wouldn’t have to wash them, regardless of their materials. The handles of the “good” steak knives were usually warped, cracked, and/or faded as a result. Wooden spoons rarely made it through a year before breaking. Cutting boards were too warped to use.
The lesson she learned, though, was NOT “Don’t put wood in the dishwasher,” but “Don’t use anything made of wood in the kitchen because it will warp or crack within a few months.”
In my own kitchen, I have a set of wooden cooking utensils, wooden-handled knives (both steak and chef), and my favorite wooden cutting board. And I’ve had some of them for more than a decade with no usability problems. Everything wooden is washed by hand.