Honestly, no one says "do not do damage" in English. I think a more appropriate translation would be something more like "don't cause (any) damage" or "don't cause (any) harm".
Do not do damage is not correct .It's a simplistic translation from Hebrew.
"Don't do any damage" sounds natural enough to me.
To do damage = to damage, so a non-annoying translation would be: Do not damage!
Yeah, but damage is a transitive verb in English. It demands an object. I wanted to say "Don't damage anything!" Though anything here is completely a filler word, it is at least natural English.
I see this as a label on a fragile object. The object will be understood.
Why damages in plural?
Well, in legal language there is the Aramaic plural נְזִיקִין like in דִּינֵי־נְזִיקִין (tort). Outside of common law you may use the common plural of segolates, i.e. נְזָקִים.
אַל תַּעֲשׂוּ נֶזֶק