Are Monster Spawners flammable?

No it's not. I was in a dungeon and I tried to surround it with lava. All it did was to stop the spawning. But the block stayed.

From the Minecraft Wiki:

• TNT, bookshelves, leaves, carpets, wool, fences, coal blocks, vines, tall grass, wood, wood planks, wood slabs and wood stairs are flammable. If flammable blocks are close to lava they can catch on fire, although the mechanics are not the same as fire spreading. Non-flammable blocks are unaffected by this effect, and do not spread active fire.
• The “embers” or “fireballs” which fly out of lava are purely decorative and do not cause fires or damage to entities. When it rains on lava, the rate at which the black "ember" particles appear increases dramatically.

NO, mob spawners CANNOT be destroyed by lava or flames. You can destroy them using TNT, your hand, pickaxes, swords, and axes. However, you can place torches to stop the monsters from spawning. Just make sure that you have plentiful torches and weapons.

I hope this answer helped! :)

No, flames or lava cannot destroy mob spawners. They can be only be destroyed using different types of weapons including: your own fist, Pickaxes, Swords, and Axes. However, you can stop the mobs from spawning by placing lava around the spawner. The lava will create enough light so that nothing can appear there.