The most obvious problem in the Middle East caused in part by WWI is the Israeli-Palestinian problem.
After WWI, the Ottoman Empire was broken up and the area that is now Israel was give to the British to govern under a League of Nations mandate. During their time in power, the British issued the Balfour Declaration, stating their intention to help create a "national home" for the Jews in Palestine. This helped to promote the Zionist movement in which Jews moved to what is now Israel. This process culminated in 1948 with the creation of the State of Israel.
In this way, WWI led to the eventual creation of the State of Israel and the resultant problems between Israel and the Palestinians.