The biggest problem I'm aware of that you could "directly" attribute to the Mongols is some of the desertification in the Iranian plateau and Mesopotamia as a result of the destruction (whether intentional or by neglect) of the incredibly delicate irrigation systems in some parts of that region.

Indirectly you could make a case for the mongols causing some intellectual shifts that 800 years later we might describe as causing some trouble in the region.

In broad strokes though? No. "The problems the Middle East faces today", things like internecine sectarian conflict, do not date from and are not attributable to the Mongol conquests. The common periodization that the Islamic "Golden Age" ended with the sack of Baghdad in 1258 belies the fact that the problems of the post-1258 era have very little to do with contemporary problems and ignore, for example, the Timurid renaissance or the flourishing of the Ottoman empire and the Pax Ottomana.

To lay the problems of the 20th and 21st century Middle East at the feet of the Mongols strikes me as entirely untenable.