Original tweeter here. :)
First of all, I'm somewhat amused/shocked that my tweet is being taken so seriously! If I had known it was going to be this widely disseminated, I would have spent more than 30 seconds writing it!
Thiago Silva is correct to point out that "static" and "dynamic" more accurately describe type checking, rather than type systems. In fact, it isn't really accurate to say that a language is statically or dynamically typed, either. Rather, a language has a type system, and an implementation of that language might enforce the type system using static checking, or dynamic checking, or both, or neither (though that would not be a very appealing language implementation!).
As it happens, there are certain type systems (or features of type systems) which are more amenable to static checking, and there are certain type systems (or features of type systems) which are more amenable to dynamic checking. For example, if your language allows you to specify in the text of a program that a particular value must always be an array of integers, then it's reasonably straightforward to write a static checker to verify that property. Conversely, if your language has subtyping, and if it permits downcasting, then it's reasonably straightforward to check the validity of a downcast at runtime, but extremely difficult to do so at compile time.
What I really meant by my tweet is simply that the vast majority of language implementations perform some amount of dynamic type checking. Or, equivalently, the vast majority of languages have some features that are difficult (if not impossible) to check statically. Downcasting is one example. Other examples include arithmetic overflow, array bounds checking, and null checking. Some of these can be statically checked in some circumstances, but by and large, you'd be hard-pressed to find a language implementation that doesn't do any checking at runtime.
This is not a bad thing. It's just an observation that there are many interesting properties that we would like our languages to enforce, and that we don't really know how to check statically. And it's a reminder that distinctions like "static types" versus "dynamic types" are not nearly as clear-cut as some people would have you believe. :)
One final note: the terms "strong" and "weak" aren't really used in the programming language research community, and they don't really have a consistent meaning. In general, I've found that when someone says that a language has "strong typing" and some other language has "weak typing", they're really saying that their favorite language (the one with "strong typing") prevents them from making some mistake that the other language (the one with "weak typing") doesn't -- or conversely, that their favorite language (the one with "weak typing") allows them to do some cool thing that the other language (the one with "strong typing") does not.