The ex-Chelsea striker, 38, was absent from the Saputo Stadium where the Impact claimed a 2-2 draw to seal a Major League Soccer play-off spot.
Montreal had earlier said he had a back injury to explain his omission.
But Biello said: "He didn't accept that he would come off the bench and, in the end, he didn't want to be in the 18."
The coach added: "There was tightness in his back, yes, but he was capable of playing. The organisation will have to handle this."
Drogba, who left Chelsea in July 2015, has been used as a substitute in two of the Impact's past four games.
He has scored 10 goals in 21 appearances this season.
Didier Drogba refused to play for Montreal Impact after being told he would not be starting against Toronto, according to coach Mauro Biello.