The England international, who scored his spot-kick in the shootout after the match ended 1-1, appeared to be upset at something said after the game and climbed into the crowd.
Video footage showed Dier climbing over seats and becoming involved in a fracas before he was restrained by security guards.
It is believed his brother was in the section where the incident happened.
"I think Eric did what we professionals cannot do but probably every one of us would do," Tottenham manager Jose Mourinho told reporters. (Reporting by Martyn Herman, editing by Ed Osmond)