Electrical wiring question - interconnect question
There may be other ways, but I'd use simple relay logic.

Three separate relays, switched contacts wired in parallel, connected to power for damper and damper.

Each relay has a coil that controls the relay - those are connected (individually - one relay coil per circuit) to the switched circuits.

Any one circuit being on will open the damper. Two or three circuits on will leave the damper open. When the last circuit is shut down, the damper closes.