~てしまう conjugation
I have learnt that the verb しまう can be used after the ~て-conjugation of a verb, normally in the past tense (~てしまった or ちゃった) in order to signify that something has been done or completed, often with negative connotations.

I have seen some other conjugations though, and I really can't make sense of what they're supposed to mean.

Example:
ママのココア飲んじゃえ

This is from a tweet and is very casual from what I can tell (no を particle, "ママ"). If I had to guess, I've really no idea what this means, I can tell it's the imperative form of 飲んじゃう, but that doesn't make any sense to me.

Question

~てしまう conjugation
I have learnt that  the verb しまう can be used after the ~て-conjugation of a verb, normally in the past tense (~てしまった or ちゃった) in order to signify that something has been done or completed, often with negative connotations.

I have seen some other conjugations though, and I really can't make sense of what they're supposed to mean.

Example:
ママのココア飲んじゃえ

This is from a tweet and is very casual from what I can tell (no を particle, "ママ"). If I had to guess, I've really no idea what this means, I can tell it's the imperative form of 飲んじゃう, but that doesn't make any sense to me.

Answer

～てしまう and ～でしまう become ～ちゃう and ～じゃう respectively in colloquial speech, ～ちまう and ～じまう　are also possible if you want to sound like a middle-aged man. ～てしまう usually means that you regret the action, show embarrassment or disappointment or that it happened by accident, but it also means, as you said, that something is completely done, meaning, for example, 本を読んでしまう to finish reading a book. In your sentence it's the meaning you quoted: to finish. ''Finish drinking mom's cocoa!'' or ''Drink up mom's cocoa!''.