Dier was filmed during the argument after climbing into the stands and chasing the supporter who made the alleged remarks following Spurs' penalty shoot-out loss to the Canaries on Wednesday night.
When video emerged of the incident, it intially wasn't clear why Dier was aggravated, but Mourinho explained why his player was emotional during a post-match interview.
The Portuguese coach stated that he did not support the England international's actions.
"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," Mourinho said.
"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."
He continued: "This person insulted Eric, the younger brother was not happy with the situation and Eric was not happy. [But] we as professionals cannot do what he did."
Spurs were bundled out of the FA Cup on penalties after the match finished 1-1 at the end of extra time.
Erik Lamela, Gedson Fernandes and Troy Parrott all missed their spot-kicks as Canaries goalkeeper Tim Krul proved to be the hero during the shoot-out.
Tottenham had dominated most of the match during the 120 minutes and had gone ahead early on through Jan Vertonghen.
But a mistake from Spurs goalkeeper Michel Vorm allowed Josip Drmic to prod home the equaliser from close range in the 78th minute.
Norwich will now play the winner of Thursday's night clash between Derby and Manchester United in the FA Cup quarter-finals.
Tottenham need to lick their wounds and bounce back with crucial Premier League and Champions League matches on the horizon.