Nothing.
It certainly gets harder for Kevin Durant and company, but the short-term ramifications don't mean much.
In the long-term Russell Westbrook's health is vital to the success of the Thunder, and there is no doubt that his presence makes it easier on his teammates, but he'll need to be 100% for the Thunder in the playoffs for them to advance as seen last year. This current injury will help facilitate the current growth of backup point guard Reggie Jackson and shooting guard Jeremy Lamb.
Both Jackson and Lamb have kept Oklahoma City at the top of the league in a season where many pundits (including myself) thought they would take a step back this year after two successive seasons of losing their scoring sixth-men in James Harden in '11-'12 and Kevin Martin in '12'-13.
Jackson's experience from last year's playoffs has seemed to pay off as he is averaging career highs in minutes, points, rebounds and assists. His PER has risen 15% up to 16.4 and he's added a reliable three-point shot. It's been Lamb though, who's unexpected play this season has provided a boon. Averaging around 10 points per game and hitting at a high percentage with around 40% from downtown, Lamb has been a potent scorer off the bench.
Along with more minutes for Jackson and Lamb, Derek Fisher becomes the full time backup, and should serve the that second unit well. Of course, this also means more shots for Kevin Durant. If you can criticize anything about Westbrook's game, it is his tendency to shoot, a lot (and his turnovers). I shutter when I see box scores where Durant shoots 10-16 and Westbrook is 8-23 in a loss (which is rare). While the shooting line doesn't tell the whole story, I've never seen a game where Durant struggled to get his shot off whenever he wants so I wonder why this is such a common occurrence. Durant struggled a little when Westbrook was out before, but with Jackson's and Lamb's improvement I'd look for Durant to go off several times in his absence.
All the Thunder need to do is win the games they should in Westbrook's absence, and they are too good to have the wheels come off. They are experienced enough that home court throughout the playoffs isn't the most important thing, health is.