What you've been told here is not accurate.
When you drive from Jerusalem to the Dead Sea, the road does cross through the Palestinian territory. You will be driving among Palestinian cars and past Palestinian cities villages, including Jericho. You will also drive past Israeli settlements established in that part of the Palestinian territory. You will have to cross checkpoints at the beginning and at the end of the road, when you leave and then reenter Israel, so take your passport with you and all the documentation of the car as well.
The road is currently controlled by the Israeli army. As a result, Israeli rental companies authorize their customers to drive on that road.
For any other travel in the Palestinian territory, you need to rent a car from East Jerusalem, as only these rentals allow you to drive in both Israel and the Palestinian territory.