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.
Orogeny (the building of mountains) is the result of converging plates, but the composition of the resulting mountain range will vary depending on tectonic setting:
The subduction of an oceanic plate under a continental plate will create a volcanic arc (such as the Andes). This is called an accretionary orogen. Partial melting of the mantle wedge above the subducting plate will generate magma, thus igneous rocks intruding the crust (or erupting at the surface).
This tectonic regime will lead to a collisional orogen. If the crust gets thick enough, the lower crust will partially melt, generating granite plutons. Also, part of the oceanic plate that once existed between the continents can be uplifted onto the crust. That's how today you can find gabbros and pillow lavas in the Alps at 3,000 m a.s.l., while they were formed at an oceanic ridge 2,500 meters below sea level!
So in any case there will be a certain amount of igneous rocks in the core of a given mountain range, even if they might not be exhumed yet.
Addressing is every volcano a mountain:
Nope. Some are actually depressions called calderas, when they have erupted and the resulting emptying of the magma chamber caused the roof to collapse, or if they erupted only once, leaving behind a maar. Examples: Yellowstone, or Phlegrean fields are calderas, volcanic Eifel has "Maare" (German plural of Maar).
There are shield volcanos, calderas, lava domes, ...