Could also mean to be stuck with a task you didn't want. 'Don't just drop it on me!' would mean something like 'Don't leave me to do this job by myself.'

Hit me with it - is give me the news you said you had Drop it on me may be a variation.

We say 'lay it in me' not 'drop it on me'. It means 'tell me the news' good or bad 'Don't drop it in my lap' means that you don't want the responsibility of something somebody it trying to give you.