The Canadian, Norwegian and Filipino men are being held by Abu Sayyaf, the Islamist militant separatists who last week beheaded Canadian John Ridsdel.
In the video, the hostages say if the demands are not met "we will be executed like our friend John".
Abu Sayyaf has previously demanded a multi-million dollar ransom.
Mr Ridsdel was kidnapped from a marina near the city of Davao last September along with another Canadian, Robert Hall, his Filipina partner Marites Flor, and Kjartan Sekkingstad, a Norwegian.
They were taken to an Abu Sayyaf stronghold of the remote island of Jolo where Mr Ridsdel was killed on 25 April after a ransom deadline passed.
The Philippines and Canadian government have said they will not give in to ransom demands. The Philippines has also launched a military operation against the militant group.
The new video, reported on Tuesday by the SITE Intelligence Group which monitors jihadist media, showed the three hostages with six gunmen standing behind them.
A masked militant warns Canada and the Philippines that the three remaining hostages would be killed "if you procrastinate once again".
Mr Hall is shown saying the governments were being ordered to "meet the demand" of the kidnappers, without giving further details.
He also asked the Philippines government to "stop shooting at us and trying to kill us. These guys are going to do a good job of that."
Mr Sekkingstad says that "if the demand is not met we will be executed like our friend John was a few days ago".
Ms Flor is seen pleading with several Philippines officials and candidates in the upcoming national election, saying "we want to be freed alive", the AFP news agency reports.
Abu Sayyaf is a fragmented but violent militant group with its roots in the Islamist separatist insurgency in the southern Philippines. Several of its factions have aligned themselves with the so-called Islamic State.
It has repeatedly taken hostages over the years but has often released them in exchange for ransoms.
On Sunday, the group released 10 Indonesian sailors they had been holding for five weeks.
It is still holding several captives, including a group of eight Malaysians and Indonesians seized from boats and a Dutch birdwatcher taken in 2012.