Fofana, who was sent off in the 72nd minute after receiving a second yellow card for a challenge on James Ward-Prowse, shared screenshots of messages he received on Instagram after the match.
"2026, it’s still the same thing, nothing changes," the Frenchman wrote on Instagram. "These people are never punished. You create big campaigns against racism, but nobody actually does anything."
Chelsea condemned the abuse on their official website.
"Such behaviour is completely unacceptable and runs counter to the values of the game and everything we stand for as a club. There is no room for racism," they said in a statement.
"We stand unequivocally with Wes. He has our full support, as do all our players, who are too often forced to endure this hatred simply for doing their job.
"We will work with the relevant authorities and platforms in identifying the perpetrators and take the strongest possible action."
Mejbri, who was fouled for the first of the two yellow cards that led to Fofana’s dismissal, also posted the messages he received on social media.
"Educate yourself and your kids," he wrote in an Instagram story.
Burnley backed the Tunisian in a statement, saying there was no space for racism at the club.
"There is no place for this in our society and we condemn it unreservedly," they said on their website.
"The club continues to be unequivocal in its stance - we have a zero-tolerance approach to any form of discrimination.
"The club has reported the post to Instagram’s parent company, Meta, and expects strong support from them, together with the Premier League and the police, and will work to ensure that the individual responsible is identified and investigated."
The draw moved Chelsea into fourth place on goal difference ahead of Michael Carrick’s Manchester United, who face Everton on Monday and could reclaim the position with a win.