The video, shot in the south Beirut suburb of Ghobeiry, shows the dogs lying on their sides near a bowl of food shaking uncontrollably and frothing at the mouth.
A municipal employee is then seen picking up their bodies and throwing them on the back of a pickup.
Welfare organisation Animals Lebanon, which posted the video, said it was seeking action from the government.
"This sickening level of abuse, torture, suffering and complete disregard for life and law is absolutely shocking," it said.
"We are communicating now with the minister of interior (Nohad Machnouk) to condemn such action and officially notify all municipalities that this is illegal and unacceptable."
Ghobeiry municipality said that it had suspended several staff members and opened a disciplinary investigation but denied any responsibility.
"This is an isolated villainous and reprehensible act carried out on the personal initiative of a number of health department staff," it said.
But the interior minister, who is responsible for overseeing local government, said that he had ordered an inquiry into the actions of the municipality.
President Michel Aoun posted a photograph of himself with his pet dog and a reminder that, under new legislation he signed into law in August, any action that causes suffering to animals is punishable by a fine of up to $13,000 (11,000 euros).
"Stray dogs may pose danger to people in several ways but the methods to resolve these cases are also numerous," he said.
"These methods are definitely not what we have witnessed on television and social media channels especially after the new animal protection law was signed earlier this year."
The leader of Lebanon's Druze community, Walid Jumblatt, posted a photograph of himself with his dog Oscar, describing the killing of strays as a "crime".
"If there is one creature to remind us of our humanity and to teach us faithfulness, love, loyalty and tenderness, it's the dog," the veteran lawmaker tweeted.
It is not the first time this year that there has been an outcry over the treatment of animals in Lebanon.
In January, the culling of hundreds of seagulls deemed a threat to flights at Beirut airport drew a torrent of criticism when activists posted pictures of the dead birds strewn on the ground after being shot by hunters.
