Wordorigins.org

The term eighty-six is restaurant/bar slang for an item that is out of stock or a customer that is to be denied service. The origin is obscure, but it seems likely that the number has no significance; it is simply part of a larger numbering scheme used by waiters and soda-jerks.

Walters and Hopkins Burlesque from 1926-35 contains this exchange, which appears to be a use eighty-six in the sense of denying a customer service, although this is not certain:

Waiter...If you need any Scotch or gin, sir—...My number is Eighty Six...Skid...Yeah. Eighty Six. I know. (Waiter exits R. Skid draws enormous flask from pocket.)