Forest were angered at being denied three penalty claims in the 2-0 loss and minutes after the game issued a statement on social media platform X accusing Video Assistant Referee (VAR) Stuart Attwell of being a Luton Town fan.
At the time, Forest were involved in a relegation battle with Luton but they eventually finished 17th while Luton went down. Forest said they would appeal the FA's decision.
"The club denied that the comments posted on social media imply bias and/or question the integrity of the match officials and/or the video assistant referee and/or bring the game into disrepute," the FA said in a statement.
"An independent Regulatory Commission found the charge to be proven following a hearing and imposed the club's sanctions."
Soon after the game, Forest said on X: "Three extremely poor decisions - three penalties not given - which we simply cannot accept.
"We warned the PGMOL (Professional Game Match Officials Limited) that the VAR is a Luton fan before the game but they didn't change him. Our patience has been tested multiple times. NFFC will now consider its options."
The FA said that no other professional club had ever breached its regulations designed to control the use of social media in such a manner, describing it as a public attack on the integrity of a match official on an "unparalleled scale".
"Even the drafters of the rule could not have foreseen a breach of this nature by a professional club," it said.
They also said Forest's concerns before the match "fell well short of a request" for Attwell to recuse himself.
Forest had asked for the fine to be suspended but the FA rejected the request.
"We are particularly concerned that the FA, in its submissions, sought a sanction 'in excess of 1,000,000 pounds ($A1.9 million)'," Forest said in a statement.
"We believe that this request, along with the subsequent fine, is wholly disproportionate and the club will be appealing the decision."
