The word "shit" is rude, don't use it. I know everyone hears it all the time on TV, but in TV dramas the emotional intensity is higher than in real life.
"Don't pull that shit on me" is an angry expression.
It means "don't try to trick me" or "don't try to lie to me." "That shit" means you hear that kind of lie all the time.
"What, you haven't received payment? Don't worry, the check is in the mail."
"'The check is in the mail?' That old story? Don't pull that shit on me."
I can't give any logical explanation for the choice of the verb "pull" or the preposition "on." It's just an idiomatic expression.