Centers will be impacted. The result to airline traffic, I dunno. As stated in the media, some operations may operate with increased distances between traffic. Sectors within a center will probably be consolidated, meaning one set of eyeballs potentially watching quite a bit more traffic, hence the increased spacing. Does that mean a total system fiasco? Probably not, just some delays here and there.
Another thing to consider: most center controllers undergo background checks, since they'll potentially be involved in AF1 and troop movements. If a controller is living paycheck to paycheck and takes a 10% hit, that could suddenly impact his/her clearance. "Can't pay the bills" is an invitation to provide some 'splaining to the 3 letter agencies. The question then becomes, does a supervisor knowingly put his/her controllers at risk of losing their clearance, or will there be special dispensation since this will be a nation-wide situation.
Or do we assume that Congress will get it in gear and provide quick resolution? I'm betting on the latter (most likely another extension...).