The Ivory Coast international has until Monday to respond to the charge, the FA said in a statement. The case wil be heard by a Regulatory Commission on Monday.
Drogba reacted after he believed a coin had been thrown at him following his goal. He was seen to pick up an object and throw it into the Burnley fans behind the goal.
He was also booked by referee Keith Stroud for celebrating in front of the Burnley fans when he appeared to gesture with one finger, although the match official said he did not witness the throwing incident.
Drogba could also find himself in trouble with the police after the Metropolitan Police confirmed on Thursday that they had received a complaint from a fan and were investigating.
The FA said it had written to both clubs to request information on missiles thrown at Chelsea players from the Burnley section.
"Players should be able to play football without fearing for their personal safety. It is totally unacceptable for players to be put in danger by supporters throwing missiles onto the pitch," an FA spokesman said.
"The FA wants anyone found guilty of throwing missiles to face the strongest possible action. That includes life bans from attending football matches."
Drogba apologised for his actions immediately after the Burnley game, which Chelsea lost on penalties, although he is expected to face a ban of three matches.
Liverpool's Jamie Carragher served a three-match ban for a similar incident at Arsenal in 2002.
