Tserenbaatar Tsogbayar and Byambarinchen Bayaraa protested after officials ruled against Ganzorigiin Mandakhnaran in the bronze-medal match.
The pair have been banned by United World Wrestling from all international competition until August 2019.
Mongolia’s national body has also been fined 50,000 Swiss francs (£39,324).
Mandakhnaran was leading against his opponent, Ikhtiyor Navruzov of Uzbekistan, by seven points to six in freestyle wrestling for his weight class.
But he celebrated too soon and danced around Navruzov for the last 18 seconds. The judges penalised him for not engaging in the fight.