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.
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"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)
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