The 22-year-old admitted a Football Association charge of ‘aggravated misconduct by reference to sexual orientation’, the body said in a statement on Tuesday.
Tunnicliffe made his comment in the aftermath of Middlesbrough’s 2-0 Championship play-off final defeat to Norwich City on May 25.
Bamford played for the Riverside team in the game at Wembley in the final match of his loan spell from Stamford Bridge, during which he scored 19 goals in 44 appearances.
Tunnicliffe will miss the first two games of next season as part of his punishment and will also attend a mandatory education course.
The former Manchester United youth player began last term on loan to Blackburn and 17 times for them before making 24 more appearances for the Craven Cottage outfit.