When you visit the Muqata, the headquarter of the Palestinian National Authority, you will be requested to identify yourself at the entrance; show your passport. A policeman will escort you to the backyard, where you can see the ruins of the buildings destroyed during the military operations. As a tourist, you are not allowed to go further. If you want to take pictures, another policeman will photocopy your passport and you will be warned not to take pictures of the people working there, only buildings. However, you can take pictures of yourself if you wish.
When you visit the area 'A' territories (according to the Oslo Accords), i.e. territories under full Palestinian control, you will have to pass through a military checkpoint. Israeli soldiers will request your passport. They will not stamp it, because officially you are not crossing any international border. Israeli citizens are not allowed to visit the area 'A' territories without special permissions. The soldiers will ask you few questions such as your place of residence (or stay) in Israel, the purpose of your visit, etc. If you forget your passport, it will be very unlikely that the soldiers will allow you to enter; maybe another national ID card can be accepted, but that is the decision of the soldiers. If you go by car, stop at the indication and do not cross until a soldier indicates it. Any misunderstanding can have serious consequences.