Warner eventually continues on to the Aussie dressing room after giving de Kock a final spray.
It also shows multiple Aussie players looking back at Warner with nervous, concerned expressions as the situation threatened to boil over.
South African captain Faf du Plessis is also seen in the video without a shirt on, joining the argument alongside his teammates.
De Kock had frustrated Australia's charge to victory in the First Test on day four, batting his way to an unbeaten 81 as South Africa reached 9/293, needing another 124 runs to win the Test.
Cricket Australia has released a statement to announce it will investigate the incident.
"CA is aware of reports of an incident between players in Durban," a Cricket Australia spokesperson said on Monday.
"CA is working to establish the facts of what has occurred and will not be commenting further until that has been done."
ICC match referee Jeff Crowe was already expected to speak with Lyon and Warner about their on-field behaviour on day four.
ICC laws dictate that players must not use "language, actions or gestures which disparage or which could provoke an aggressive reaction from a batsman upon his/her dismissal".
Warner's run out of de Villiers was the big talking point out of a dramatic day of play.
Former South African cricket captain Graeme Smith has called Warner a "fool" while adding Lyon may regret his unnecessary celebration following de Villiers' run out in the Kingsmead Test.
Australia's vice-captain unleashed a spray on centurion Aiden Markram, who was involved in the run out, while Lyon dropped the ball in the direction of de Villiers who was sprawled on the turf in trying to make his ground.
"The Aussies were pretty fired up at that time pushing for a victory," Smith told cricket.com.au in Durban, where Australia need just one wicket on day five for victory.
"But he (Lyon) is an experienced cricketer.
"I think he'll probably say it was unnecessary himself.
"We've got used to Davey over the years. I think the less interest you take in him the better.
"He can be a bit of a fool at times. It's best just to let him be." Smith warned the teams not to cross the line with sledging.
"I think everyone that's played the game has hard moments and that's what makes the hundred from Markram so special," Smith said.
"He's had to work, he's showed great composure, the Aussies were all over him after that run out.
"Everyone wants to be competitive, people love watching that.
"I think there's always a line that you've got to be careful you don't cross." Australia require just one wicket to take a 1-0 lead in the four-Test series after setting South Africa 417 for victory.
Aussie Test great Simon Katich said Warner's on-field reaction to the run out was pure emotion and perfectly understandable. He said dismissing de Villiers was a massive moment in the match and understands why Warner reacted how he did.
Katich does not, however, understand or support the way Lyon responded to the de Villiers wicket.
"Once he was run out the emotions that came from David Warner were quite natural because he knew it was a big wicket," Katich told SEN Radio.
'He executed a beautiful run out. What followed with Nathan Lyon is something he'll probably regret doing. There was no malice in it, but it just didn't look good.
"What Nathan Lyon did, he'll regret putting the ball near AB de Villiers. That sort of action is something that could possibly have invoked a reaction from the batsman and that's not something the game needs at that point in time.
"It just wasn't a good look."
It came as Markram and Aussie keeper Paine both publicly declared the game was played in the right spirit during their press conferences after play on day four.
"There wasn't too much aggression," Paine said.
"We spoke to Aiden about running out their best player and one of the best players in the world.
"I think had someone had run Smithy out in our team you'd cop a fair bit of a ribbing. It was nothing aggressive. It was just reminding him of what he had just done, trying to get him off his game, the same as they do to us. It didn't work."
