mold into dishwasher
While the high temps typically used in a dishwasher will kill a lot of life and will probably suceed in doing so with this dish, no you should do something to it first.

The dishwashers instructions certainly advise you to essentially clean dishes before putting them into the dishwasher. Anytime food or other "solid matter" is left on a dish, it will go to the diswashers drain, and it may very well stay there.

You are potentially likely to add the mold (alive or dead) to the filter, the grinder, etc, and promote clogging. Not to mention a small chance of smell or more rot to occur. Why risk it?

Flush the contents of the cup if its sensible, throw the contents in the trash, rinse the contents down the drain, etc. Wipe it with a paper towel if you feel that will help.

Then dish wash away.