Answering the question - No, it would not make sense.

Currently the closest things we have to a multi-language VM are the JVM and the CLR. These aren't exactly lightweight beasts, and it would not make sense to try and embed something of this size and complexity in a browser.

Let's examine the idea that you could write a new, multilanguage VM that would be better than the existing solution.

You're behind on stability.

You're behind on complexity (way, way, behind because you're trying to generalize over multiple languages)

You're behind on adoption

So, no, it doesn't make sense.

Remember, in order to support these languages you're going to have to strip down their APIs something fierce, chopping out any parts that don't make sense in the context of a browser script. There are a huge number of design decisions to be made here, and a huge opportunity for error.

In terms of functionality, we're probably only really working with the DOM anyway, so this is really an issue of syntax and language idom, at which point it does make sense to ask, "Is this really worth it?"

Bearing in mind, the only thing we're talking about is client side scripting, because server side scripting is already available in whatever language you like. It's a relatively small programming arena and so the benefit of bringing multiple languages in is questionable.

What languages would it make sense to bring in? (Warning, subjective material follows)

Bringing in a language like C doesn't make sense because it's made for working with metal, and in a browser there isn't much metal really available.

Bringing in a language like Java doesn't make sense because the best thing about it is the APIs anyway.

Bringing in a language like Ruby or Lisp doesn't make sense because JavaScript is a powerful dynamic language very close to Scheme.

Finally, what browser maker really wants to support DOM integration for multiple languages? Each implementation will have its own specific bugs. We've already walked through fire dealing with differences between MS Javascript and Mozilla Javascript and now we want to multiply that pain five or six-fold?

It doesn't make sense.