There are very few results which allow us to know that a mathematical claim will be provable or disprovable within ZFC without actually proving or disproving it. To the best of my knowledge, the only exceptions are theories which have [quantifier elimination][1]. Few<sup>1</sup> open mathematical problems which people are interested in are of this sort, and none of the Millenium problems are. So any of the Millenium problems could be independent of ZFC (except for the Poincare conjecture, because it has been proved!)

You might be particularly interested in [Scott Aaronson's survey][2] on whether or not it is likely that $P \neq NP$ is independent of ZFC.

<sup>1</sup> Here is an example of a question which I know is decidable in ZFC, yet whether the answer is "yes" or "no" is open.

> Do there exist $44$ vectors $(u_i,
> v_i, w_i, x_i, y_i)$ in
> $\mathbb{R}^5$, each with length $1$,
> and with the dot product between each
> pair $\leq 1/2$? See [Wikipedia][3] for background.

This is the a first order question about real numbers, so it is decidable by [Tarski's theorem][4]. The analogous result for four dimensional vectors was only obtained in [2003][5]; if you can get the answer for $5$ dimensions, it should be publishable in a good journal.  I think this about as interesting a question as one can find which is definitely settleable in ZFC, yet still open. Most questions mathematicians care about are not of this form.


  [1]: http://en.wikipedia.org/wiki/Quantifier_elimination
  [2]: http://www.scottaaronson.com/papers/pnp.pdf
  [3]: http://en.wikipedia.org/wiki/Kissing_number
  [4]: http://en.wikipedia.org/wiki/Real_closed_field
  [5]: http://mr.crossref.org/iPage/?doi=10.1070%252FRM2003v058n04ABEH000651