Other mistake: When the camera changes shot to the security feed for the first time in River's bar brawl scene, you can see, on the top right, Jayne and Mal sitting down with Fanty and Mingo and the Dancer preparing to block the camera's view. This is before the gang have even stepped into the bar.
Other mistake: When the Assassin is pulling up the identity and history of Malcolm Reynolds, based upon the security video of River's bar fight, it shows that he was a volunteer for the rebellion forces, but lists his rank as Captain, while we know from several instances on the Firefly series that he was a Sergeant.
Answer: She has several mental disorders, presumably she finds shoes confining or uncomfortable. It's notable that when she does have footwear it's almost always heavy calf high boots, instead of slippers or more comfortable shoes.
Grumpy Scot
To add; She likes to dance, and being barefoot it's easier to move around.
lionhead
Something I always wondered is when she goes on the heist barefoot at the beginning of the film, none of the other characters seem to notice or care that she's in bare feet. Even her brother, Simon, didn't try to get her to wear shoes.