Can regeneration save my creatures from Turn // Burn?
Yes, you can activate an activated ability as often as you can pay its activation cost.

But you don't need to. The ability will still resolve even if the creature no longer has the ability when it resolves.

For example, say the creature in question is a Dutiful Thrull.


Your opponent casts Shock targeting your Dutiful Thrull.
In response, you activate its ability.
In response, he casts Turn.
Turn resolves: The Dutiful Thrull is now a 0/1 red Weird with no abilities.
Dutiful Thrull's ability resolves: A regeneration shield is placed on Dutiful Thrull.
Shock resolves: It deals two damage to the 0/1 Dutiful Thrull.
SBAs are performed: The creature would be destroyed, but the regen shield kicks in. Dutiful Thrull is tapped, removed from combat and all marked damage is removed.


You could activate Dutiful Thrull's ability a second time between steps 3 and 4, which will give it a second shield for later on in the turn.




  112.7a Once activated or triggered, an ability exists on the stack independently of its source. Destruction or removal of the source after that time won’t affect the ability. [...]