Jurgen Kantner was abducted from his yacht off Malaysia's Sabah state in November. His companion Sabine Merz's body was later found on the boat.
A deadline for a 30m peso (£483,000; $600,000) ransom expired on Sunday.
Mr Kantner, 70, and Ms Merz had been abducted before. They were held for 52 days in 2008 by Somali pirates and were released after a ransom was paid.
The video, reported by the SITE militancy-monitoring group, shows Mr Kantner being killed by a knife-wielding man.
Government envoy Jesus Dureza confirmed the killing.
