Wenger was angered by a decision to award Burnley a penalty late in the game on Sunday which Arsenal won 2-1.
"It is alleged that in or around the 92nd minute, he used abusive and/or insulting words towards the fourth official," the FA said in a statement.
"It is further alleged that following his dismissal from the technical area, his behaviour in remaining in the tunnel area and making physical contact with the fourth official amounted to improper conduct."
Wenger has until 1800 GMT on Thursday to respond.
