Mr Kasyanov, a former prime minister, is a high-profile critic of Russian President Vladimir Putin, in the opposition RPR-Parnas party.
Recently Mr Kadyrov called Mr Putin's critics "enemies" and "traitors".
Russian opposition politicians have described the posting as a murder threat.
The Chechen leader said Mr Kasyanov was seeking cash in Strasbourg for the opposition. "Whoever doesn't get it will get it!" he warned.
Last March Mr Kadyrov spoke out on Instagram about the assassination of opposition leader Boris Nemtsov in Moscow. That message defended one of the Chechens charged over the shooting.
A day later President Putin gave Mr Kadyrov a top award.
Mr Nemtsov was among several well-known opponents of Mr Putin murdered in the past decade.
Putin loyalist Kadyrov unleashed on Russian 'traitors'
Ramzan Kadyrov: Putin's key Chechen ally
Mr Kadyrov runs Chechnya with an iron fist - his private militia has been accused of widespread human rights abuses, including torture and assassination.
He has close ties to Mr Putin, who encouraged him to stamp out a separatist insurgency in Chechnya. Thousands of civilians died in the North Caucasus republic when Russian troops fought the rebels there in the 1990s.
The new video is the latest in a series of threatening messages from Mr Kadyrov against Kremlin critics, whom he accuses of working for the West.
It shows Mr Kasyanov talking to Vladimir Kara-Murza, a journalist who runs the pro-democracy opposition movement Open Russia. The movement was launched by exiled oligarch Mikhail Khodorkovsky.
The video was posted late on Sunday and has more than 16,000 "likes".
Mr Kasyanov says he sees it as a direct death threat and will demand a criminal investigation.
Ilya Yashin, who co-chairs RPR-Parnas with Mr Kasyanov, called the video "an open threat to murder Kasyanov".
Another opposition leader, anti-corruption campaigner Alexei Navalny, echoed that view. On Facebook, Mr Navalny said (in Russian) "there is no longer any doubt that all such statements in recent weeks and specifically this one were approved by Putin and the Kremlin, and quite probably were inspired by them too".
The BBC's Sarah Rainsford in Moscow says they key question now is whether Mr Kadyrov is acting on his own initiative.
One theory discussed on Russian social media is that he is being used by the Kremlin to intimidate its critics - particularly as growing economic problems raise the potential for protest.
Others suggest that Mr Kadyrov is becoming a dangerous loose cannon - a regional leader who believes he is untouchable, our correspondent reports.
The phrase "Whoever doesn't get it will get it!" is the title of a thriller that Mr Kadyrov says he has filmed, starring himself as a machine gun-toting all-action hero.
Kadyrov the Instagram fan - by Stephen Ennis, BBC Monitoring:
Ramzan Kadyrov is one of Russia's most popular and controversial Instagram users. Since launching his account in 2013, he has made over 6,000 posts and amassed 1.6m followers.
He promotes himself on the site as a devout Muslim and a strong leader. One recent post featured him grappling with a crocodile.
This is not the first time he has used Instagram to lash out at the Russian opposition. In May 2015, hours after the premiere of an Open Russia film about him, he posted a clip of himself firing a machine gun, with the slogan "Whoever doesn't get it will get it soon!"
The following day Open Russia co-ordinator Vladimir Kara-Murza was taken seriously ill with poisoning.
Instagram has rapidly grown in popularity in Russia in recent years and at the end of 2015 had more than 10m active users.
Chechnya's leader Ramzan Kadyrov has posted an Instagram video showing Russian opposition politician Mikhail Kasyanov in a sniper's crosshairs.