Alisson had earlier been booked for arguing with the referee, after the fourth official did not have the electronic board ready for Andoni Iraola's triple substitution when the match was level at 1-1 after about 70 minutes.
Forest restarted the game with a quick throw-in, as Neco Williams charged into the Liverpool box and Becker brought him down, leading to a penalty that put visitors Forest 2-1 ahead.
After Liverpool salvaged a draw, 33-year-old Alisson walked by the substitution board placed on the ground outside the tunnel. Videos on social media showed the keeper kicking the board, damaging the equipment.
"It was alleged that the goalkeeper acted in an improper manner around the tunnel area after the final whistle. Alisson Becker subsequently accepted the charge against him and received the standard penalty," an FA spokesperson said on X.
Liverpool, who drew their opening two league matches before beating Ipswich Town, will host Fulham on Saturday.