What lasting effects of the Six Day War are still being felt in the Middle East today ?

I will summarize it in one word : Apartheid.
Summary About 6.8 million Jewish Israelis and 6.8 million Palestinians live today between the Mediterranean Sea and Jordan River, an area encompassing Israel and the Occupied Palestinian Territory (OPT), the latter made up of the West Bank, including East Jerusalem, and the Gaza Strip. Throughout most of this area, Israel is the sole governing power; in the remainder, it exercises primary authority alongside limited Palestinian self-rule.
More than 14 million people, roughly half of them Jews and the other half Palestinians, live between the Jordan River and the Mediterranean Sea under a single rule. The common perception in public, political, legal and media discourse is that two separate regimes operate side by side in this area, separated by the Green Line. One regime, inside the borders of the sovereign State of Israel, is a permanent democracy with a population of about nine million, all Israeli citizens.