Every volcano is a mountain, but not every mountain is a volcano. Still, it strikes me that--at least for the mountain ranges I can think of in this moment--they all seem to have igneous cores. Is this generally true, or are there plenty of examples of mountain ranges with sedimentary or metamorphic cores?
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
examples of mountain ranges with sedimentary or metamorphic cores
At an elevation of 7000 metres and higher, it is all sedimentary rock. Below 7000 metres, it is metamorphic of sedimentary protolith. There are some igneous intrusions into it, but the bulk is a metamorphic schist.