Images and videos taken in the away supporters' section at Carrow Road show Dier in a heated argument as stadium security guards attempt to break up the fracas.
Spurs manager Jose Mourinho said after the match that a fan had insulted Dier's brother, but the manager didn't approve of his player's reaction.
Eric Dier lept into the stands to confront supporters following Tottenham's penalty-shootout loss at Norwich.— ESPN FC (@ESPNFC) March 4, 2020
Jose Mourinho said the fans were insulting him and his younger brother got involved.
(via @jselby123) pic.twitter.com/a1uzS71VCQ
"This person insulted Eric, this family was there. The young brother was not happy with the situation and then Eric, I repeat did what we professionals cannot do, but did something that probably we would do," Mourinho said.
"I don't think that belongs to the game. The game was a fantastic representation of the FA Cup, but I cannot run away from the question. I think Eric Dier did what we as professionals cannot do but when someone insults you and your family is involved, especially your young brother.
"If the club discipline Eric Dier I would not agree with it, but what he did was wrong."
On if the incident was also based on racist abuse toward a player, Mourinho added: "I cannot say, I don't know. I just know that the reaction of Eric was based on his brother, didn't like what was happening there. I cannot go in that direction with these details."
The FA have sought observations from Tottenham and club stewards and they will study CCTV footage as they prepare to open disciplinary proceedings over the incident.
Norwich keeper Tim Krul saved two spot kicks including the final two from Troy Parrott and Gedson Fernandes to help the Canaries oust Tottenham and advance to the quarterfinals for the first time since 1992.
Wednesday's loss and potential discipline for Dier add to Mourinho's woes after an initial revival when the Portuguese replaced Mauricio Pochettino.
Their run of defeats includes a 1-0 home loss to RB Leipzig in the Champions League, defeat away to Chelsea and a home loss to Wolverhampton Wanderers at the weekend that left Tottenham in seventh place in the Premier League.
"I am not going to say what everybody knows so this time I am not going to say that, but I have to say Steven Bergwijn, it's too much for him to play so many minutes like he is doing," Mourinho said, "I prefer to call Lucas Moura a hero, he is playing every minute of every game in every position.
"Harry Winks was completely dead. I think he's started 11-12 games in a row, we had many many players in trouble."
Information from Reuters and ESPN correspondent Mark Ogden added to this report