"Please open window" is a sign I saw on a London bus. What rule let the author omit an article? It was written right on the window, so I'm wondering why they didn't write "Please open the window".
An older term for this kind of writing is “telegraphic style.” Telegrams charged by the letter, so people would send messages as short as possible to save money. Businesses, such as banks and brokerages, would have codebooks full of abbreviations to save money on telegrams. Often, these were important instructions.
SEND MONEY would have been a typical example.
This especially became famous as a style of communications between military units, where the concern wasn’t cost, but messages were often sent in code, or by a very inconvenient mechanism like signal flag or beacons. In WWII in particular, codebreakers often worked by looking for a common word that appeared in most messages, like
EINS in German, so avoiding words like
THE could make a code harder to break.
I personally wouldn’t call that example “headlinese,” because headlines normally aren’t commands like that. But newspaper headlines were similarly terse and clipped. This was so a print newspaper could make the letters in the headline as big and catchy as possible.
This kind of sign has meaning when attached to a specific window. It does not need the article because it relates to the window it is attached to. The same for "Keep off Grass" only works on a sign "in the grass".
It is not something we would say or write in other situations.
As a native speaker that is how I interpret that aspect of the grammar; other grammarians can give their views.