Weir was dismissed for a professional foul in the last minute of the 3-0 defeat. Many observers seemed to think that the contact took place outside of the area, but it is likely that even had the referee agreed, it would still have been a professional foul and therefore a red card offence.
An Everton spokesman said: "David Moyes, along with club officials, have looked again at the incident involving David Weir and have decided not to lodge an appeal. However if the referee was to take another look at the video himself, then that would be a different matter."
Weir will now miss Everton's away game against West Ham on November 2nd.