The 28-year-old Wales worldwide striker was shown a straight red card late on in his side's Premier League victory at Turf Moor.
The Welshman came off the bench to score the victor against the Clarets but was sent off 12 minutes later for a challenge on Matt Lowton.
The 28-year-old striker will miss the Baggies' next three games through suspension unless the club's claim of wrongful dismissal to the Football Association is upheld.
A statement on West Brom's official website confirmed the outcome.
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The unsuccessful appeal means Robson-Kanu will now be suspended from Tuesday night's Carabao Cup tie at Accrington Stanley and the Premier League games against Stoke City and Brighton.
Robson-Kanu grabbed the victor for West Brom during their 1-0 victory over Burnley on Sunday but he got himself sent off later on after seemingly elbowing into Matt Lowton.
Tony Pulis' men performed a smash and grab raid at Turf Moor, thanks to substitute Hal Robson-Kanu's late victor.
The Baggies have won back-to-back games at the start of a Premier League season for the first time, but Pulis' focus is now on bolstering his squad.