looks like your tree has botrytis.

I haven't sprayed anything recently. In the past I've used Neem oil in the past but not in the last few weeks.

Any thoughts on how to treat this?

Botrytis is a result of high humidity, and poor sanitation. Sanitation is the first important step. Remove EVERY BIT of dead and infected tissue from the plant, and from the soil surface. Discard all dead and infected plant parts completely out of the entire residence. Keep the humidity as low as possible around the tree. Water the tree only in the morning and then set the tree in the sun to quickly dry the plant. Doing so vigilantly should take care of it.

The tree has been outside for 6 months and in the last couple of weeks has only been watered once every 2 days. We also very rarely get rain here in Austin. It's usually hot and sunny. I don't quite understand how this happened in the first place.

This was the tree this morning. Does it still look like botrytis? There is not dust; just dry dead leaves.

Could it be sunburn?

Would it be possible to get sunburn in November after being in the sun all summer without issue?