Arsenal had defeated Burnley on Sunday, thanks to a last-gasp penalty goal from Alexis Sanchez.
A 10-man Arsenal were leading until the 93rd minute when a foul from Francis Coquelin was awarded in favour of Burnley and they equalised from the spot.
This left Wenger in fumes as agitatedly approached fourth official Anthony Taylor and allegedly bad-mouthed him.
Taylor gave him the marching orders but Wenger stood at the entrance of ground. Taylor once again approached the French manager and this an angry Wenger pushed him.
FA's official statement read:
"Arsene Wenger has been charged for misconduct following Arsenal's game against Burnley on Sunday [22 January 2017].
"It is alleged that in or around the 92nd minute, he used abusive and/or insulting words towards the fourth official.
"It is further alleged that following his dismissal from the technical area, his behavior in remaining in the tunnel area and making physical contact with the fourth official amounted to improper conduct.
"He has until 6 pm on 26 January 2017 to respond to the charge."
Arsene Wenger had apologised in front of the media just after the game thus it is highly unlikely that he would appeal against the decision of the FA.