0759 GMT November 20, 2019
Defender Francis, 31, was dismissed for a challenge on midfielder Aaron Ramsey as the Gunners came from 3-0 down to rescue a point at the Vitality Stadium, BBC reported.
Cherries boss Eddie Howe said it was a "harsh" decision by referee Michael Oliver to send off Francis, before Olivier Giroud equalized in added time.
"It was a foul but I don't think it was a sending-off," said Howe.
The club expects to hear the outcome of its appeal by the end of the week.
If the appeal is unsuccessful, Francis will miss Saturday's FA Cup third-round tie with Millwall as well as Premier League games against Hull City and Watford.