Do cyclocross races permanently damage the grass they are run on?
I've observed a number of cyclocross races in public parks in the area I live. These typically include paved areas, grassy areas and some dirt. If there's been rain recently (or raining during the race), the grass along the race course looks like it's been turned to mud. If you look more closely, you can see that the individual blades of grass are all intact, they're just bent over and muddy. A few months later, you can't even tell where the race course was anymore.

I believe it depends on the type of grass. For well-established and regularly mowed grass, such as is common in parks and for playing sports on, the damage is temporary and cosmetic. The ruts settle out on their own.

It seems like areas of slope that cause the racers to dig in and slip around more can get more damage.

Here's a photo essay of some cyclocross races the day of and several weeks later: 
http://api.ning.com/files/zdPAeE6CK7J69K-gbRX4ru1-SAay5M6pkU6QqXD5QiNJnWx4-gdFn9oV0w47vypkoo67sVWzznYDLM4RwIDwgGVxfQP8Ivn0/AfterEffects.pdf (from http://www.ukcyclocross.com/profiles/blogs/after-effects-of-a-cyclocross).

Cyclocross races are generally run during a season that grass is dormant. It may not fully recover to looking nice until a few months later when the sunny/warm season starts up again.