You can put a toilet seat in a dishwasher to wash the toilet seat and disinfect it but I would not do so because it's not really sanitary to put things from your toilet into something such as your dishwasher that you wash your dishes in.
Just like it's not sanitary to use your toilet plunger that you plunged your toilet with in your sink to unclog your sink with.
I would just fill the bathtub with some hot water and add some ammonia or other disinfectant to the bathtub water to clean the toilet seat.
Or just use some Clorox Disinfecting Wipes on the toilet seat or use some bleach in the bathtub water and clean the toilet seat in it.
But I would not place the toilet seat in the dishwasher because it's not really sanitary to do so.