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.
"Up to the last moment, many sectors, including the armed forces, exhausted all efforts to save his life. We all tried our best but to no avail," he said.
Abu Sayyaf is one of the smallest and most violent jihadist groups in the southern Philippines, known for its brutality, including beheadings.
It has pledged allegiance to so-called Islamic State and has carried out kidnappings of foreigners and Filipinos. Some have been released for ransoms but a number are still being held.
A Philippine police report said Mr Kantner had been killed in the Indanan area of southern Sulu province on Sunday afternoon.
However, no body has yet been recovered.
The Philippine military had carried out air strikes on Abu Sayyaf positions in Sulu over the weekend as the deadline approached.
After the German couple's yacht, the Rockall, was found off Laparan Island in Sulu province on 7 November, Abu Sayyaf had issued an audio message saying it was responsible for the abduction.
Ms Merz was apparently killed as she tried to resist the militants.
After the 2008 abduction off Somalia, Mr Kantner returned to the self-declared republic of Somaliland to collect his yacht, saying: "My boat is my life and I don't want to lose her... I don't care about pirates and governments."
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