Will spells that exile enchantments also exile enchantment artifacts?
Yes, any card that can exile an enchantment can exile an enchantment artifact. This is because an enchantment artifact is both an enchantment as well as an artifact.


  205.2b Some objects have more than one card type (for example, an artifact creature). Such objects satisfy the criteria for any effect that applies to any of their card types.


There's also this nugget of information buried in the Basic Rulebook, to help illustrate this relationship to new players.


  Artifact creatures are both artifacts and creatures. [...]  An artifact creature can be affected by anything that affects artifacts, as well as anything that affects creatures.


Trivia: The quoted piece of information was present in the M14 Basic Rulebook, but you won't find it anywhere in the M15 Basic Rulebook.

Examples

Erase can exile Whip of Erebos, because Whip of Erebos is an enchantment, and Erase targets enchantments. Being an artifact does not make this any less true.

Dryad Arbor would be affected by Urborg, Tomb of Yawgmoth, because Dryad Arbor is a land.

Dryad Arbor would also be affected by Anger of the Gods, because Dryad Arbor is a creature.

A Darksteel Citadel enchanted with Ensoul Artifact with an Enchanted Evening in play would be would be affected by anything that affects enchantments, creatures, artifacts, or lands.