I'm none-too-sure if the West Wing will survive this. First off Sorkin really was the driving force behind the show. If you watch anything he's written, whether it's Sports Night, or The American President, you'll recognize in it the same great dialogue that makes The West Wing a good show. Without Sorkin, you lose the dialogue, and I'm no longer certain the show will be as entertaining.
Furthermore, we know that some of the actors on The West Wing have scrambled for money beforehand, so it's likely to happen again. The "ensemble" cast includes many high-paid regulars, so the show is an expensive one to maintain, yet it routinely ranks below ER and all 3 L&O incarnations on the ratings scale. Since the L&O series all have smaller casts, NBC is making significantly more money on these shows than on the West Wing. Don't get me wrong, The West Wing almost always finishes in the top 20 shows every week, I'm not saying it's without an audience.
Still, given the West Wing's high production cost, even a small dip in viewership (due, of course, to the lack of witty dialogue after Sorkin leaves) may be enough to make NBC "take the show in a new direction".