Armenia needed to win their final Group B qualifier to finish second and make the play-offs but they lost 2-1 on Tuesday after their goalkeeper Roman Berezovsky was sent off by Spanish referee Eduardo Gonzalez after 26 minutes.
Berezovsky was dismissed for handball outside his area but TV replays clearly showed he chested the ball clear.
FFA chief Ruben Airapetyan told reporters the Armenians want European football's governing body to cancel Berezovsky's red card rather than punish the referee.
TV replays also showed the referee allowed play to continue after it appeared Ireland striker Simon Cox did handle the ball moments before Berezovsky was wrongly adjudged to have done so.
He later sent off Ireland's Kevin Doyle after the striker receieved two yellow cards.
Airapetyan dismissed any conspiracy theory after some Armenian fans suggested UEFA was trying to give Ireland an advantage to balance out France striker Thierry Henry's infamous handball that cost them the chance of reaching the 2010 World Cup finals.
Henry handled the ball in the build-up to France's winning goal against Ireland in the playoffs but the offence went unpunished.
"Trust me, I would never think or believe that it would be possible for UEFA to give such instructions to the referee," Airapetyan said.
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