Is it permitted to use AI to write divrei Torah or prepare a drasha?

Be careful here. A davar Torah is not only information to be delivered; it is the fruit of your own wrestling with the text. When you let a machine do the wrestling, you may produce something polished and say something true, yet skip the very avodah that was supposed to change you in the process.

I would not call it forbidden, but I would call it a temptation. Use it the way you would use a concordance or a search of the meforshim — to find sources, to check a memory, to surface a question you had not considered. Then close it, sit with the sugya yourself, and let the words you finally speak be ones you have actually thought.