What happens if
Phyrexian Metamorph copies something that was a creature, but then ceases to become a creature? For example, a Gideon that turns back into a planeswalker, or an enchantment animated by
Starfield of Nyx and then Starfield is removed?
I'm guessing that in the first case Phyrexian Metamorph dies (since it doesn't copy counters, which should include loyalty counters) while in the second, it turns into an enchantment?
With Starfield of Nyx it would become the Enchantment as the Enchantment was printed. Starfield would then make it a creature and it would be an Artifact Enchantment Creature. If you ever lost enchantments and went below 5 then Metamorph would become just the Artifact Enchantment and lose it's creature status.
Gideon. He would become Gideon as the card is printed. He would however, have no loyalty counters. He'd die immediately unless you had something that gives Planeswalkers Loyalty Counters when they ETB.
Ruling in the Gatherer Page - Except for also being an artifact, Phyrexian Metamorph copies exactly what was printed on the original permanent and nothing more (unless that creature is itself copying something or is a token; see below). It doesn’t copy whether that permanent is tapped or untapped, whether it has any counters on it or Auras attached to it, or any noncopy effects that have changed its power, toughness, types, color, and so on.
You are correct in the Starfield example,
Phyrexian Metamorph will copy the animated enchantment and if Starfield leaves or the condition is no longer met, it will become just an artifact enchantment.
In the planeswalker example, it's a different story. With
Gideon animated as a creature, he is a "Planeswalker Creature - Gideon Human Soldier". If
Phyrexian Metamorph is played and
Gideon is selected, it will enter the battlefield as a copy of
Gideon. A non-animated planeswalker. It will have loyalty counters placed on him, just like any planeswalker has when it enters the battlefield. Before you can do anything else, state-based effects are checked, and because only one planeswalker with the name Gideon can be under your control, you will have to choose which of them you wish to move to the graveyard. See this
link for more info.
[edit] This assumes everything above is under your control. If your opponent has a planeswalker which has somehow been animated as a creature, you could clone that and both planeswalkers will remain on the battlefield as they are controlled by different players.