I have to admit that there's no reason at all why anyone should ever use this word.
It's an exclamation of contempt or disagreement, and, yes, it's ludicrously old-fashioned. Nowadays the middle-aged might say nonsense! (if they were being polite) and the young might say yeah, right.
In pleading for pish I can only point out that you're saving a syllable...
...unless you use the form pish posh! which means the same thing.
But that's obviously ridiculous.
Pish-tushery is a term used to describe ridiculously elaborate language in historical fiction. You know the sort of thing, I'm sure: it's often when an author is determined not to waste any of her research.
'Pish, sir!' exclaimed Lady Charlotte, as she opened the patch box that she held on her hooped skirt which had been woven in Spitalfields by the newly immigrant Huguenots. 'La! How shocking! Do you not know that Lord Pumpkin's gout has necessitated his removal to Bath in his chariot? Tush!'
Good grief...I've spent twenty years being careful to avoid that sort of thing, and the effort of concocting that paragraph has nearly killed me.
Word To Use Today: pish. I've no idea where this word came from, and neither has my dictionary. Still, it'd be sad to see it perish.