Cars are built to withstand a lot more punishment than an autocross can deliver. Unless your drivinga a rust bucket, any car will do fine. Sure I can see if you are running skinnys up front a Mickey T's out back, then you might want to skip your next auto-x venue. But, any car in stock form or better can compete just fine.
As far as the danger....very little of it. No one will pull out infront of you, no obsticles. A safe are area to race. Its certainly something any "car guy" or girl should try at least once. It really gives you a better understanding of your car's and your own driving limits.
The only thing I've ever seen go wrong was at FSF. Some guy in a convertible BMW 323 broke a rear suspension compenent and his wheel went up under his wheel well. He was able to drive it off course. That is rare though.
Your tires will get wear too, but not as bad as you think. I autocrossed a set of soft Yokohama AVS Intermediates, and they were my daily driver rubber too. I got close to 70K miles out em!!!
Bottom libne...come and try it. You will be glad you did. If people don't want to race, let 'em be. Everyone is entitled to an opinion.