Kruphix, God of Horizons makes it so mana, instead of emptying, turns colorless. On a similar note, Omnath, Locus of Mana, makes it so green mana does not empty between phases. How do these effects work with restricted mana, like mana generated with Food Chain or Shrine of the Forsaken Gods?
It's still the same mana —it simply changes/loses/gains color— so the restrictions remain.
From Kruphix's Gather rulings:
4/26/2014 If unused mana in your mana pool has any restrictions or riders associated with it (for example, if it was produced by Cavern of Souls), those restrictions or riders will remain associated with that mana when it becomes colorless.
From Omnath's rulings:
3/1/2010 If a green mana you add to your mana pool has certain restrictions or riders associated with it (for example, if it was produced by Ancient Ziggurat), they’ll apply to that mana no matter when you spend it.