Enchantment Hate -- Well, of course. Back to Nature is your worst nightmare, but you might be able to recover depending on when in the game it gets used. If you expect Back to Nature to come out of your opponent's sideboard, you should probably bring in counter spells from yours to try and block it.
As for targeted removal, exile effects hurt more than destroy effects since you can bring enchantments back from the grave. Starfield of Nyx is automatically the best target for targeted enchantment hate since it can bring other cards back. Try to protect it more than other enchantments for this same reason.
Board Wipes -- Or at least, board wipes controlled by your opponent. By turning your enchantments into creatures, any board wipe your opponent uses becomes stronger than normal and will almost certainly clear your board. Starfield of Nyx doesn't turn itself into a creature, but a second copy will. Be very wary about placing two copies of Starfield of Nyx onto the field at the same time, even knowing that they can bring back more cards each turn. If all your copies hit the graveyard, there are no built-in ways to get them back.
Also, this deck does run Supreme Verdict even though board wipes are a weakness. Supreme Verdict is here to clear your opponents' creatures before your enchantments become creatures. Once Starfield of Nyx becomes active, think carefully before using it--it might be correct at some point if you have lots of auras and can bring cards back right away, but it MUST hurt your opponent more than it hurts you to be worth it.
Graveyard Hate -- Since a lot of power comes from using Starfield of Nyx to bring enchantments back from the graveyard, things that are strong at keeping graveyards empty (such as Rest in Peace) can be painful to deal with. A one-time graveyard nuke like Tormod's Crypt isn't so bad since you can continue to use your graveyard after it's done, but if you're going up against lingering graveyard hate, you'll need to focus on being able to play from your hand and/or library. If you can keep up counterspells for the graveyard bombs, that's great. If you can't and they prove difficult to remove, you'll want to work harder to protect your existing cards, such as through use of Privileged Position. Starfield of Nyx isn't useless in this situation since it can still animate your enchantments, but it won't be gaining you advantage, so keep this in mind when deciding when it get it out as opposed to other cards.