If Finder of Paths Unseen causes Heart of the Wildfire to add/move Presence, destroying Invaders, does Finder need to pay for "Responsibilities to the Dead"? What if Finder destroys some of Volcano's Presence? Or if Finder moves Dahan during Dahan Insurrection?

If Finder of Paths Unseen causes Heart of the Wildfire to add/move Presence, destroying Invaders, does Finder need to pay for "Responsibilities to the Dead"? What if Finder destroys some of Volcano's Presence? Or if Finder moves Dahan during Dahan Insurrection?

No for Wildfire and Dahan Insurrection. Yes for Volcano, assuming the damage from destroying Volcano's Presence destroys 1 or more Invaders/Dahan.
When adding/moving Wildfire's Presence generally or moving Dahan during Dahan Insurrection, Finder's action itself did not cause any destruction. It "directly triggers" the destruction (via Wildfire's "Blazing Presence" special rule or the "Constant Raiding" scenario rule), but it does so by adding/moving Presence/Dahan, not by moving the destroyed Invaders. (Earlier printings of Wildfire incorrectly said "when" instead of "after" for the Special Rule; see the errata / clarifications on triggered actions.) This remains true even when a Power like "Traveler's Boon" moves both the Presence/Dahan and the destroyed Invader – while it did move the Invader and did directly trigger the destruction of the Invader, it didn't directly trigger the destruction of the Invader by moving it.
"Collapse in a Blast of Lava and Steam" is phrased as a modifier when Volcano's Presence is destroyed, it modifies the current action to also do 1 Damage. If that action is Finder's, it's Finder dealing the damage. (This would have worked more thematically if "Collapse in a Blast of Lava and Steam" had been a triggered action, since then it would have been Volcano dealing the damage.)