The Ivory Coast international is under fire after throwing a coin back into the crowd while celebrating his 27th-minute goal against Burnley in the Carling Cup fourth-round tie at Stamford Bridge.
Investigations into the incident have been launched by the Metropolitan Police and the FA.
Although Drogba issued a post-match apology for his behaviour, it may not save him from further action.
A Metropolitan Police spokesman said: "I can confirm we are investigating the incident. An inquiry is being conducted by the Football Unit at Fulham police station. No-one has been arrested."
An FA spokesman added: "We are investigating. That's all I can say until we have had a chance to look into it further."
Taylor said, "It's there for all to see, he's done something that's against the rules required of players, and a similar situation resulted in a three-match ban for Jamie Carragher," he said.
"So it would be very difficult to avoid an accusation that they (the FA) are inconsistent if they don't follow that line."
"For Didier Drogba, he's been injured, he's come back, he's got a good goal but he got wrapped up in the situation, and it is a fact for footballers that they're not supposed to celebrate in front of opposition supporters," Taylor added on Sky Sports News.
"This is the second coin-throwing incident we've had now in the last few weeks, which is a matter of some concern, but the last thing that players need to do is respond in that way.
"In fairness to the lad, he knows he's done wrong, he apologised immediately after the game, but this happened in the heat of the moment and unfortunately there will be consequences now."