The drivetrain failures aren't likely to be much of a problem on a standard car, but you should still expect things to wear out more quickly. Remember that a race car will generally have engine and transmissions rebuilt a lot more frequently than a roda car (I used to have an annual rebuild of the Ultima transmission, and it was only doing about 1000 miles between rebuilds).
On a modified car (which is quite likely to be the one running with slicks) you're going to see much higher wear on transmission components. Basically, if you have an extra 50 ft lb of torque over standard from your engine, and you're seeing wheelspin on road tyres, but not with the slicks, that extra 50 ft lb of torque is trying to pull your transmission to bits. The more modified, the more wear and tear on the transmission.
Just make sure that everything is 100% before you take it on the track, and get it checked a lot more frequently than usual.
On another note. It's always astonished me that people will take sick cars out on a track day. The number of people I've heard say that they're going to get their car serviced after a track day, and not before.