Koscielny was dismissed in the first half of Sunday's 3-1 victory at Emirates Stadium after a late challenge on Enner Valencia.
Arsenal launched an appeal against the decision of referee Michael Oliver but the Football Association rejected the club's plea on Tuesday.
Koscielny will therefore serve a three-match ban, the first game of which will be Saturday's FA Cup final against Premier League champions Chelsea at Wembley.
Laurent Koscielny will serve a three-match suspension after his wrongful dismissal claim was rejected: https://t.co/VDPrW3jT3H— The FA (@FA) May 23, 2017
Arsenal are facing a defensive crisis for the final, with Arsene Wenger also expected to be without Gabriel after the centre-back was substituted due to injury against Everton.
Per Mertesacker came off the bench to make his first Premier League appearance of the season, while Shkodran Mustafi missed the Everton game with a head injury sustained in Arsenal's victory over Sunderland.
Alexis Sanchez was also substituted in the second half of the Everton match, with the Chilean forward limping out of the game with 22 minutes to play.