The Italian was sent off for leaving his technical area and walking down the touchline towards the end of the game and there was an altercation between both sets of players and staff members at full-time.
Sarri subsequently refused to speak to the media after the game, with assistant boss Gianfranco Zola taking on those duties.
Zola explained that the tactician felt "frustrated" by being sent off and "offended" by something said to him from the Burnley bench.
The former Chelsea player insisted that his compatriot only left his area to assist the officials and tell his players to fall in line, and was therefore not pleased that he was sent to the stands as a result.
The Blues endured a blow to their hopes of a top-four finish as they drew 2-2 at Stamford Bridge against the Clarets, and their manager could now face a sanction.
A hectic start to the game saw Jeff Hendrick volley home the opener inside the first 10 minutes, but quick-fire responses from N'Golo Kante and Gonzalo Higuain before the 15-minute mark saw the Blues come from behind to take the lead.
However, Ashley Barnes restored parity 10 minutes later and Chelsea were unable to force home a winner. There were accusations after the game of time-wasting by Sean Dyche's side, with Tom Heaton picking up a yellow card in the first half for taking too long over his goal-kicks.
Regardless, the result takes Chelsea temporarily fourth as both Arsenal and Manchester United lost - against Crystal Palace and Everton, respectively - but just one point ahead of the Gunners, having played an extra game.
Sarri remains under pressure after a disappointing first season in charge at Stamford Bridge and a potential touchline ban could be a blow to the Blues' hopes of clinching Champions League football next season.