The Birmingham manager was left furious after Linington reportedly mocked Robinson for being in a losing position following a challenge between the defender and Burnley forward Danny Ings which led to a goal for the visitors.
"There was an incident involving Danny Ings and Paul Robinson," Clark told reporters. "I thought it was a foul by Ings.
"They scored from the resultant free kick. Paul remonstrated with the referee, who gave Paul a yellow card, but he mocked him about the scoreline at the time. This is totally unacceptable.
"When my captain is fighting for the club and the supporters are backing my team and when I am working as hard as I can to get the results to remain in the league, getting mocked by an official is something I am not going to accept."
The draw leaves Birmingham 19th in the Championship.