Does this mean that the action only occurs on your next turn after the unit (or support, etc..) comes into play or does it occur every turn that the unit is in play.

For example, I play the Bloodthirster which has the Forced ability "After your turn begins, each player must sacrifice a unit in the corresponding zone".

DUring the beginning of my next turn we both have to sacrifice a creature, that's clear enough. But what about the turn after that?

Is it just a one time occurence or does it occur at the start of each of my turns as long as the creature is in play?

As long as the BT is in play, sacrificing happens at the start of your turn. A Forced activates each time the trigger is met (turn begins in the BT's case).