Smith was sent off in the latter stages of Saturday's fifth-round tie at stadium:mk for leading with an arm in a challenge on Rory Delap.
The 32-year-old's ban was increased from three to four games because it was his second dismissal of the season.
Dons boss Karl Robinson said after the game that he would be appealing against the decision.
However, in a statement, the Milton Keynes club said unsuccessful attempts earlier this season to have red cards shown to Stephen Gleeson and Antony Kay overturned had convinced Robinson to not lodge an appeal.
"While the manager does not agree with the red card, following unsuccessful appeals to overturn red cards issued to Stephen Gleeson and Antony Kay earlier in the season, the decision was made not to enter the FA appeal process," the statement read.
Smith will miss the League One trips to Carlisle and Portsmouth as well as home fixtures against Preston and Doncaster.
MK Dons have had seven red cards this season - joint highest in the Football League along with Notts County.