According to the very Wikipedia article you cited, there is no such thing as "runtime type inference". What you described as such is a way of implementing dynamic type checking: checking at runtime that the types of the various parts of an expression actually match. You should actually read your sources.
So, "type checking" isn't always performed at compile time.
Type checking is a little different, and refers to some safety measures done at compile time that theoretically guard against certain types of errors. Scripting languages don't do them, and don't miss out on much programming correctness as far as I can tell.
Full run time type inference in C++ is about having a container of something -- let's say void pointers that just point to memory locations, and figuring out what sort of thing they're pointing at. Or similarly, you have a base-class pointer and want to figure out which derived version of the object is being pointed at.
I replied to you above, but I'll reply to your specific example here. Your example is really not how PL people talk about type inference. However, you're right that you wouldn't call this example type-checking either... it would probably be a fuzzier term such as "reflection" or "introspection". But certainly not type inference. Do you understand what the wikipedia article you linked is about? Because it's NOT about introspecting the type of a contained object within a container. It's about statically making proofs of the type of a variable based on its usage.
A quick read of this page and its top link strongly suggest that runtime type inference is an optimization designed to speed up the interpretation of dynamically typed programs. It sounds very useful for JIT compilation (to make appropriate code specializations). It's also likely too complex to be implemented in simple interpreters (like Lua's).
Note that this term isn't very widely used: we are already in second position at your link.
So, unlike ML-style compile-time type inference, runtime type inference is implementation specific. Your earlier statement "Scripting languages have run-time type-inference built in everywhere" is actually ill-typed. However, if you had said "checking" instead of "inference", your sentence would have been correct.
Hence my "Err, by "runtime type inference", you actually mean runtime type checking, right?"
Type checking is what you say when the compiler makes sure that you've used all your types correctly, and spits out an error when you don't. It's there to prevent you from trying to assign an int to a string, for example.
"Runtime type checking" is the same thing at runtime. Errors and exceptions get thrown, your program stops. So no, that's not what we're talking about.
There is such a thing, but it has a different meaning than how I believe you used it.
You said "scripting languages have run-time type inference built in everywhere". I don't think that's true for most scripting languages. Inference refers to determining facts which were not explicitly provided, and usually implies a statically typed programming language where inference is done at compilation time. "Runtime type inference" might be used for, e.g., Python's Psyco project, where the just-in-time compiler infers properties about a particular variable (such that it's always an integer) and can therefore compile out boxing, unboxing, runtime type checks and so on.
Speaking of runtime type-checking, I inferred that that's what you meant when you said "runtime type inference". Type-checking IS something that happens at runtime almost everywhere in dynamic languages. It is also more closely associated with RTTI -- a dynamic_cast in C++ would not be termed type inference (the previous and new types are known), but a type check does occur at runtime.
The pipe dream you are talking about has a tiny hope of coming true here: http://www.bitc-lang.org/
EDIT: replying to thras, below.
According to the very Wikipedia article you cited, there is no such thing as "runtime type inference". What you described as such is a way of implementing dynamic type checking: checking at runtime that the types of the various parts of an expression actually match. You should actually read your sources.
So, "type checking" isn't always performed at compile time.