Are the cores of every mountain range igneous?
Many mountain ranges do not have igneous cores. The front ranges of the Rocky Mountains in British Columbia are created by thrust faults that push sedimentary strata up to form the mountains. The driving force for this motion is a subduction zone located 100's of kilometres to the west. I believe that the Himalayas mountains are formed in a similar fashion.  

See the following Wikipedia article: Geology of the Rocky Mountains