The incident occurred on 4 March in the fifth-round tie following a penalty shootout in which Dier scored, but Spurs eventually lost.
There was no physical exchange and the matter was not pursued by the Metropolitan Police after a brief investigation.
However his claim was dismissed by the English football authority, and Dier will miss four of Spurs' final five Premier League matches, against Bournemouth, Arsenal, Newcastle and Leicester, although he will be available for their final match of the season, away to Crystal Palace.
"Eric Dier has been suspended for four matches with immediate effect, fined £40,000 and warned as to his future conduct following a breach of FA Rule E3".
Spurs manager Jose Mourinho said the club were unlikely to appeal the decision. "We all know how it works with these FA decisions". I am not going to answer, if I answer I am going to be in trouble and I don't want to be.
Just after the incident on the night, Mourinho said: 'This person insulted Eric, the family was there, the young brother was not happy with the situation then Eric did what we professionals can not do but we probably would do.
"You have access if you want to the record of successful appeals and unsuccessful appeals".
'When somebody insults you and your family is there and they get involved with the person insulting him I think Eric Dier did what we professionals can not do but probably what everyone of us would do.
"I think he had a unusual last season and also a very odd current season", added Mourinho.
"I think he is going to be an important player for us".
Tottenham's Eric Dier will miss this weekend's North London Derby clash with Arsenal due to suspension after he picked up a four-game suspension.
Mourinho has confirmed Dele Alli will miss the trip to the Vitality Stadium after failing to recover from a hamstring problem, which ruled him out of Monday's 1-0 win over Everton.
"He had a little history with hamstring injuries last season so he has to be careful".
Mourinho added of the problem: "Small, but not ready for tomorrow".