When I google "drop litter", only results from British websites appear. What is/are typical phrases used in America against littering? Is it simply "do not litter" or is the verb "drop" also used?
No, Americans don't use "drop litter". Instead, you'll find signs that just say "no littering", such as this one:
There are also signs that say "do not litter":
Images from Recycle Reminders