If the target Mountain is an illegal target when Peak Eruption tries to resolve, it won't resolve and none of its effects will happen. No damage will be dealt. However, if Peak Eruption resolves and the Mountain isn't destroyed (perhaps because it has indestructible), damage will be dealt.

Peak Eruption can target any land with the subtype Mountain, not just those named Mountain. It can't target a land that doesn't have that subtype, even if the land can tap for {R}.