What it means
Sling your hook is a proper old-school British brush-off. It means go away, clear off, do one. It’s got a salty, slightly grumpy flavour to it, the sort of thing you’d say when someone’s overstayed their welcome or is hanging about being a pain. Not usually savage, but definitely not an invite to stay.
Usage examples
"The landlord told the troublemakers to sling their hook."
"If you are only here to complain, you can sling your hook."
"You’ve had your pint, caused your scene, now sling your hook."
"Oi, if you’re just here to stir it, sling your hook."
"He kept mouthing off at the back, so the gaffer told him to sling his hook."
Where it comes from
It comes from old British nautical talk. Hook referred to an anchor, so to sling your hook meant haul it up and leave. By the 1800s it had sailed out of dockside speech and into everyday British chat, where it stuck as a colourful way to tell someone to bugger off.