Does palestinian nationalism still have influence over contemporary politics of Israel, Palestinian and Arab neighbors?
I would have to say yes. There is much heated debate about the relationship between Palestine and Israel, but there are some big facts that help to explain how the two groups function and why there might be conflict.
First, Palestine is more of a region than a nation - so nationalism might not be the right term for how those who are Palestinian feel about their homeland. There have been many different governments and leaders of the area, and the more in-depth discussion of what those were can be found on britannica.com, or even wikipedia - (http://en.wikipedia.org/wiki/State_of_Palestine). However, the overarching ideology of Palestine, very simplified, is that those of Palestinian descent, or citizens of the Palestine area have a right to live where they wish. However, in the late 1940s, Palestine included what is now Israel. The area was under British Mandate, and after the events of WWII, the United Nations agreed that the Jewish people should be given a piece of land near their ancestral home; despite the fact that the area was already populated. Many people who are sympathetic to the idea of Palestinian "nationalism" feel that the Jewish settlers, whose descendants are the majority of Israelis, were given the land unfairly and pushed out the residents of the area without much care as to what happened to them.
As many of those people who have Palestinian ancestry or are currently living in Palestine can also be considered Arab, and are mostly Muslim (though there are Christians in the area as well), they share many traits with the surrounding countries of Jordan, Syria, Egypt, Lebanon, Saudi Arabia, Iraq, and most other countries in the Middle East. As such, many people in those countries feel that they have an obligation to help their Palestinian neighbors and feel a great deal of empathy and compassion towards them. There is still a great deal of resentment for the way that Israel's statehood came about, and despite many attempts at peace talks and negotiations, the state of Israel and the peoples of Palestine (mostly the West Bank and Gaza Strip - for a map, here is an uncredited image that gives you an idea: http://fasttimesinpalestine.files.wordpress.com/2009/10/four-panel-map.jpg) remain separate and hostile to each other.
Without question, this hostility does not apply to all people living in the area. However, the threat of war and death is a very real, day-to-day fact for the people of the area, and as such the tensions between those who side with Palestine and those who side with Israel are very high.
To sum up my answer, (tl:dr), yes. Palestinian nationalism still has a great deal of importance for the world, and certainly dictates some of the politics of Israel, the Palestinian National Authority, Egypt, the United States, and the United Nations. It is a very complex situation, and is one that is quick to cause arguments. I would recommend a good deal of research before committing an opinion about the situation, as the history goes back to biblical times.