Referee Darren Drysdale brandished a straight red card to Marney for his challenge on Paul Robinson an hour into the contest at Turf Moor.
The score was 1-1 at the time of Marney's dismissal, and Burnley's 10 men held out until the final minute when Marlon King struck a dramatic winner to lift Birmingham eight points clear of the relegation zone.
Burnley boss Sean Dyche was furious with Drysdale's decision to send off Marney, claiming the tackle was not dangerous and carried no intent to injure Robinson.
After reviewing the incident on video, Dyche insisted there was enough evidence to have the decision overturned.
He said: "We will be appealing. We felt it was the wrong decision at the time and since then we have looked at the incident again in the cold light of day to confirm our belief.
"We've also sought a few opinions from other people that confirm our thoughts, so we will put our faith in the appeals process in the belief we can get the right outcome and have the red card rescinded."