> E.g. if a function takes an interface as parameter type, you can create a dummy implementation which throw exceptions for any operation. So basically you have the same problem as you describe for dynamic languages

That's not at all the same problem, because here you're intentionally writing code that throws exceptions for some reason? Also, many languages, such as Java will not allow you to throw a checked exception that's not declared on the interface method - such issues will be statically checked and caught - thanks for proving my point ;)

> of valid inputs and an infinite number of invalid inputs. But this is the same for statically typed languages, and the type-checker will not help you, since a valid JSON string and an invalid JSON string look exactly same to the type checker.

Not at all, in a statically typed language the JSON would typically be deserialized into a structured type with all the benefits of type checking, validation and usage of the resulting values. Of course it's possible to just use a JSON string directly, but that's not idiomatic and generally not the way it's done in quality codebases.