2 reasons things die in dishwashers:
- The detergents
- The heat
I don't think you have to worrry about the detergents, they're a lot tamer than industrial kitchen dishwasher chemicals, and tamer still then most photo chemicals (both acid and alkali).
The heat may be a problem, depending on how hot your dishwasher gets, but they don't tend to go much above 80C on the most intense cycle, mine washes down at 40C on 'eco' setting, and trays definitely handle that.
If in doubt, sacrifice one and see?