I have to disagree with many of the comments here, kit should never be a limiting factor, in my opinion that's a very negative attitude to have. There are ways and means of shooting anything regardless of the kit you have. I've shot a few 24hr races with equipment that would seemingly be laughed out of the circuit (Micro 4/3s, for example), whilst the results were not spectacular, I was pleased with them and a number were picked up by teams/drivers etc.
In many ways the basics are no different to normal shooting, it's just more challenging.
- Find the brightest areas you can, anywhere with spot lights, buildings nearby with plenty of ambient light etc, anything to make life that little bit easier.
- Nail your panning technique, this is critical in my opinion (no quick easy fix to that though!), a smooth steady pan at slow shutter speeds can open up much darker areas to you, or buy you a lot of extra time whilst the sun is setting/rising etc.
- Be creative, just experiment, it's a highly unusual situation to be shooting in, so just try stuff, it's amazing all the different patterns and abstracts you can get during the night.
- Try manual focus if you haven't already, it's not as difficult as it seems and it will reduce the workload of your camera/lens significantly. If I'm parallel to the track I tend to focus on the far white line marking the edge of the track, then you just need to pan and fire.
You really need to work with whatever light you can find, one of my favourite things is to keep an eye out for cars following others closely, the headlights throw enough light out to illuminate the car ahead...
Silverstone Britcar 24hr 2011 by
Harry_S, on Flickr
That was taken with a Panasonic G3 and 45mm 1.8 (through a fence), but most of the shots later in the set below were taken with a 100-300mm f4-5.6, which on a 3 generations old M4/3 body is hardly the quickest (or brightest) of lens.
http://www.flickr.com/photos/harry_s/sets/72157607412120603/
If you're looking to spend a bit I think a 50mm 1.8 (less than £100) would be a far better investment than hiring a lens, should get some lovely night shots of all sorts of things with one of those.