This is a difficult one to explain - it has a lot of references within references within references.
In movies and tv shows in the 40s and 50s, middle class and upper class women would often wear pearl necklaces. In those movies and shows, when they were distressed or upset or sad or shocked, the actresses would grab their necklaces in a fist. It was used so often in movies, that it is considered overacting - and silly. When you say that someone is "clutching their pearls" it means they're acting more distraught than they really are (usually to prove to other people how upset they are).
So, when one character says "drop your pearls," he's telling the other character to "stop clutching your pearls." Or in more words, he's saying "I know you're not as _____ as you're pretending to be, so stop pretending to be ____." It is closer to "stop pretending to be so surprised."