"Do not go online."
Incorrect. Literally it can mean that in some contexts, but 不要 has the same function as 別 here. That is, it signifies the negative imperative/command.
Further, unlike Japanese and some other languages, you shouldn't assume the objects and subjects are dropped (your parenthesis). Chinese is slightly more strict.
Actually 不要 can also mean don't, as in "i don't want you to..." But with a less possessive feeling than in English.
How is this relevant to the Health Group Lesson? Maybe the internet is bad for your health.