Does a dies trigger still go on the stack if the creature is destroyed and replaced by a regeneration shield?

Regeneration is a replacement effect. When a regeneration shield is applied, it replaces the event of the creature being destroyed with a different sequence: the creature is tapped, all damage is removed from it, and it is removed from combat (CR 701.15b). Because destruction is replaced, the creature never actually goes to the graveyard as part of this event.

A 'dies' trigger is defined as a triggered ability that looks for a permanent going from the battlefield to the graveyard (CR 700.4). Since regeneration prevents the creature from ever leaving the battlefield, the condition for the trigger is never met, and nothing goes on the stack.

It is important to understand that regeneration is not 'saving' the creature after it dies — it intercepts the destruction event entirely before the creature can move zones (CR 614.1a). The creature simply stays on the battlefield in a tapped state.

Concrete example: Your opponent controls a Grizzly Bears with a regeneration shield and you cast Lightning Bolt targeting it. Lightning Bolt deals 3 damage, which is lethal, so the game tries to destroy Grizzly Bears. The regeneration shield applies, replacing the destruction: Grizzly Bears stays on the battlefield, tapped, with damage removed. No 'dies' trigger from any source fires, because Grizzly Bears never moved to the graveyard.

If instead the creature were exiled or sacrificed, regeneration would not apply (CR 701.15c), the creature would go to the graveyard or exile respectively, and any relevant triggers would function normally.