Mileage varies. I’ve seen “86 [thing]” written on whiteboards more often than not, grammatically speaking.
Also, a single cherry is the norm, perched decoratively atop the whipped cream. So “86 the cherry” would have been clear, and they could maybe get away with “86 cherry” according to you, but “86 cherries” might as well be “69 cherries.” You wouldn’t expect that to mean mutual oral sex.
You’re right, that would have been the “correct” way, with the “the.” When spoken it’s almost always said, or in the past tense like “cherries are 86’d.”
Of course, “no cherry” is leagues superior when you’re the customer, I mean really. He was just asking for a high ass employee who fully knew to just do it because they think it’s absolutely hilarious (and that would have been the right move lol.)
The other commenter is also right, the whiteboards in the kitchen always leave out the “the,” but that is a shorthand on a shorthand. They also probably write like “86 B.O” for “We are currently out of black olives,” and you don’t want to know how they abbreviate Jalapeños. The whiteboard is not a reliable source in that respect, it’s almost code, or like a Chef’s Cant.