Drogba had just netted his first goal of the season on 27 minutes and celebrated in front of the visiting supporters in the Shed stand.
He then threw back into the crowd a coin that was hurled at him.
He also made a single finger gesture to the taunting supporters.
Drogba has since apologised for the incident but the matter will not end there.
The FA will look at the referee's report - Keith Stroud booked the Ivory Coast striker and the police are studying footage.
Drogba, if found guilty by the FA will get a mandatory three-match ban - the same sentence handed out to Jamie Carragher in 2002 during a similer incident against Arsenal in the FA Cup at Highbury.
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