The 30-year-old was shown a straight red card for serious foul play following his early challenge on Millwall striker Theo Robinson and will now serve a three-match ban.
Burnley manager Brian Laws said: "We're disappointed with the outcome. We felt the decision was harsh, but we have to accept the verdict and move on.
"We again showed on Saturday the strength we have within the squad and I know that whoever benefits from Clarke's absence will come in and do a good job for us when we start again after the international break."
Burnley felt the challenge by Carlisle had not endangered Robinson and did not merit a sending off, but the appeal was rejected by an FA commission, which upheld the referee Keith Stroud's decision.
Clarke will miss the Clarets' Championship fixtures against Sheffield United, Barnsley and Reading later this month.