I am not a database expert and have no formal computer science background, so bear with me. I want to know the kinds of real world negative things that can happen
It's actually not correct that MongoDB is not ACID-compliant. On the contrary, MongoDB is ACID-compilant at the document level.
Any update to a single document is
What MongoDB doesn't have is transactions -- that is, multiple-document updates that can be rolled back and are ACID-compliant.
Note that you can build transactions on top of the ACID-compliant updates to a single document, by using two-phase commit.