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.

Just to add to what the previous members have said: The choice of main verb in this phrase is an analogy with other similar collocations, such as 'pull a stunt', 'pull a trick' 'pull a fast one' 'pull one over on someone'. The choice of noun ? Sheer lack of imagination.

MAYBE... A dictionary reminds me that one of the meanings of the word "pull" is "(Slang) To draw out (a weapon) in readiness for use: 'pull a gun;' 'pulled a knife on me.'" "Pull" obviously makes sense because you are in fact pulling the weapon from a holster or sheath, and it carries the idea of surprise or trickery, and the proposition "on" is sometimes used to indicate the target of violence or disrespect.

As you can guess from the vocabulary used, it is not a polite expression. The rough meaning is "don't ever try and trick me"