Three of his games in charge came against Sheffield United, 1-1 draws in the league and FA Cup before the Blades eventually won the Cup replay 3-2 on Tuesday.
QPR, watched by co-owners Briatore and Bernie Ecclestone, fell 3-0 behind in front of their home supporters, with Hart booed when he brought off midfielder Alejandro Faurlin, to replace him with Adel Taarabt.
Hart's other games were a 2-1 home win over Bristol City and a 3-0 defeat at Ipswich, both in the league.
QPR lie 10th in the Championship, four points outside the play-off places.
Under Briatore, managers John Gregory, Luigi De Canio, Iain Dowie, Paulo Sousa, Magilton and Hart have departed while Harford, Gareth Ainsworth (twice) and youth coaches Steve Gallen and Marc Bircham have had spells in caretaker charge.
Former Charlton and West Ham manager Alan Curbishley has already been linked with the job, along with ex-Reading boss Steve Coppell.
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