There is a great post about this exact issue on Quora. To answer your questions...
Since October 2000, a military warrant of a major general in the IDF central region command (Maj. Gen. Itzhak Eitan at the time) forbids Israelis from entering into the Palestinian Authority A areas (as designated in the Oslo accords).
The warrant is very short (1 page, Hebrew) and states that the area is "closed": Israeli citizens as recognised by Israeli law, are forbidden from entering, and required to immediately leave if they are in that area. People who are "not citizens of the area, who have a standing visa to be in Israel" are not allowed to enter area A as well. ("מי שאינו תושב האיזור ובידו אשרת כניסה בתוקף לישראל" which may refer to foreign nationals, the language is vague)
So legally speaking the situation is not entirely clear. The military order itself is on shaky legal grounds and it's not being systematically enforced:
The warrant was tested in Israeli courts and ruled to be one that can
be enforced (i.e soldiers may prohibit entry for an Israeli citizen),
though in April 2013 a civil court ruled that civilians failing to
comply with it are not to be arrested (after the fact).
On the ground, de facto, Israeli Arabs are not stopped at the
checkpoints and generally speaking, usually allowed to enter area A
without hindrance. Also tourists / foreign nationals are usually
allowed in. So, almost always, only Israeli Jews are turned back.
This means you should be okay, regardless of your current visa in Israel. If stopped by Israeli military personnel apologize and try again at another time. Also see our related question on visiting Palestine as an Israeli citizen.
P.S. A rough Google Translate of the linked military order from October 2000 can be found here.