The 36-year-old was making his eighth appearance of the season for Antonio Conte's side and first since November but was dismissed for a professional foul on Lee Angol in the 67th minute.
Speaking after the match, Conte suggested a covering defensive run made by Branislav Ivanovic meant Terry was not denying a clear goalscoring opportunity, although the FA have rejected this claim and any other possible mitigation.
"John Terry will serve a one-match suspension with immediate effect after his wrongful dismissal claim was unsuccessful, following an Independent Regulatory Commission hearing," read a statement on the organisation's official website.
"The Chelsea defender was sent off for denying an obvious goal-scoring opportunity during the FA Cup third-round tie against Peterborough United on Sunday."
Terry will miss Sunday's trip to face Premier League champion Leicester City, although he has not started in the top flight for Conte's league leader since the 2-2 draw at Swansea City in September.