Centre-half Collins was shown a straight red card for a challenge on Latics forward Matt Smith after an hour of the third-round tie at the City Ground last weekend.
Reds boss Alex McLeish felt at the time that the decision was harsh and the Scot remains convinced of Collins' innocence.
However, he thinks the video evidence of the incident is not clear enough to support their case.
"We have decided not to appeal," McLeish told the media ahead of Saturday's home game against Peterborough.
"Not because we are not entirely convinced that Collins was not the perpetrator, but purely because the pictures were not clear enough.
"We did not want to risk getting done for making a frivolous appeal.
"We were punished by a harsh decision during the game and now there is a further punishment, in the form of a one-match ban for Danny, at a time when we are short in that position."