Putting wooden spatulas in the dishwasher shortens their life, but it doesn't destroy them straight away. I don't feel bad at all about putting cheap wooden cookware in the dishwasher. And, of course, if you're using non-stick pans, worrying about black plastic is a mote/beam kind of situation.
I'm not even sure it does that. My cheapo wood spoons/turners/spatulas get dishwashed ~every time they're used, and they're 10? years old at this point. One finally cracked a few years ago but the rest are going strong.
