i don't know of it as an expression, but i think they just meant it could also be interpreted as "don't eat the illness," which sounds like telling someone not to get sick
it's like a half-literal half-metaphorical expression because most common illnesses are spread through your mouth/spit
"Don't drink the bottle!" should be accepted or it should be the main answer.
That's incorrect in English. Closest would be "Don't drink from the bottle" but that's not what it says in Korean, does it?
Korean translates 먹다 to "drink" or "eat" depending on the context, but it still has to be meaningful in English.