My advice is pretty much the same as the others' here - read your policy carefully!
Most policies have a clause that voids your coverage if you are "racing" (e.g., engaged in a "prearranged speed contest" in my AAA policy). (Of course, if you and the Ferrari in the next lane get crazy on the street and you wreck, that is fully covered!?!?)
My interpretation of this is that if/when something goes wrong during practice - particularly when I have a student in the car and we aren't comparing times - then I am covered, as there is no contest in progress. However, should I have an off-track excursion during the timed runs, my coverage is voided, and any damage would come out of my pocket - I wouldn't even try to file the insurance claim.
CAVEAT - I am not an attorney and am not claiming to be an expert in insurance policies, the law, or anything else in this area. The opinions expressed here are just that - opinions. If you want legal answers, consult a lawyer.
I have been told that there are policies where people were able to get letters from their insurance agent stating that they were covered in an AX event. However, I have never seen such a letter, so this may be another case of urban legends. (I am talking about standard insurance, not insurance specifically for track events.)
Of course, the real solution is to drive within your limits!

("A man's got to know his limitations." Dirty Harry)